K11a171

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K11a170.gif

K11a170

K11a172.gif

K11a172

Contents

K11a171.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a171 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X18,5,19,6 X20,8,21,7 X2,10,3,9 X22,11,1,12 X6,14,7,13 X8,15,9,16 X12,18,13,17 X14,19,15,20 X16,22,17,21
Gauss code 1, -5, 2, -1, 3, -7, 4, -8, 5, -2, 6, -9, 7, -10, 8, -11, 9, -3, 10, -4, 11, -6
Dowker-Thistlethwaite code 4 10 18 20 2 22 6 8 12 14 16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a171 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a171/ThurstonBennequinNumber
Hyperbolic Volume 18.6712
A-Polynomial See Data:K11a171/A-polynomial

[edit Notes for K11a171's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -2

[edit Notes for K11a171's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+7 t^3-21 t^2+39 t-47+39 t^{-1} -21 t^{-2} +7 t^{-3} - t^{-4}
Conway polynomial -z^8-z^6+z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 183, 2 }
Jones polynomial -q^8+5 q^7-12 q^6+19 q^5-26 q^4+30 q^3-29 q^2+26 q-18+11 q^{-1} -5 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -4 z^6 a^{-2} +2 z^6 a^{-4} +z^6-5 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} +2 z^4+z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} +4 a^{-2} -2 a^{-4} -1
Kauffman polynomial (db, data sources) 3 z^{10} a^{-2} +3 z^{10} a^{-4} +9 z^9 a^{-1} +20 z^9 a^{-3} +11 z^9 a^{-5} +23 z^8 a^{-2} +29 z^8 a^{-4} +16 z^8 a^{-6} +10 z^8+5 a z^7-7 z^7 a^{-1} -23 z^7 a^{-3} +z^7 a^{-5} +12 z^7 a^{-7} +a^2 z^6-65 z^6 a^{-2} -71 z^6 a^{-4} -23 z^6 a^{-6} +5 z^6 a^{-8} -21 z^6-9 a z^5-15 z^5 a^{-1} -18 z^5 a^{-3} -28 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -a^2 z^4+47 z^4 a^{-2} +47 z^4 a^{-4} +12 z^4 a^{-6} -3 z^4 a^{-8} +14 z^4+4 a z^3+14 z^3 a^{-1} +21 z^3 a^{-3} +17 z^3 a^{-5} +6 z^3 a^{-7} -6 z^2 a^{-2} -8 z^2 a^{-4} -4 z^2 a^{-6} -2 z^2-z a^{-1} -z a^{-3} -z a^{-5} -z a^{-7} -4 a^{-2} -2 a^{-4} -1
The A2 invariant q^8-3 q^6+3 q^4-3 q^2-1+6 q^{-2} -4 q^{-4} +8 q^{-6} -2 q^{-8} + q^{-10} + q^{-12} -6 q^{-14} +4 q^{-16} -3 q^{-18} +2 q^{-22} - q^{-24}
The G2 invariant q^{46}-4 q^{44}+11 q^{42}-24 q^{40}+36 q^{38}-43 q^{36}+30 q^{34}+20 q^{32}-100 q^{30}+210 q^{28}-303 q^{26}+316 q^{24}-204 q^{22}-78 q^{20}+476 q^{18}-859 q^{16}+1062 q^{14}-921 q^{12}+376 q^{10}+457 q^8-1316 q^6+1850 q^4-1784 q^2+1060+97 q^{-2} -1270 q^{-4} +1969 q^{-6} -1893 q^{-8} +1078 q^{-10} +148 q^{-12} -1230 q^{-14} +1686 q^{-16} -1303 q^{-18} +249 q^{-20} +1016 q^{-22} -1900 q^{-24} +2001 q^{-26} -1197 q^{-28} -222 q^{-30} +1715 q^{-32} -2695 q^{-34} +2772 q^{-36} -1870 q^{-38} +304 q^{-40} +1366 q^{-42} -2538 q^{-44} +2797 q^{-46} -2076 q^{-48} +684 q^{-50} +801 q^{-52} -1800 q^{-54} +1931 q^{-56} -1212 q^{-58} +9 q^{-60} +1116 q^{-62} -1648 q^{-64} +1352 q^{-66} -394 q^{-68} -811 q^{-70} +1726 q^{-72} -1967 q^{-74} +1468 q^{-76} -447 q^{-78} -674 q^{-80} +1480 q^{-82} -1735 q^{-84} +1427 q^{-86} -747 q^{-88} -13 q^{-90} +601 q^{-92} -883 q^{-94} +847 q^{-96} -591 q^{-98} +270 q^{-100} +17 q^{-102} -196 q^{-104} +252 q^{-106} -230 q^{-108} +152 q^{-110} -71 q^{-112} +14 q^{-114} +22 q^{-116} -31 q^{-118} +28 q^{-120} -20 q^{-122} +10 q^{-124} -4 q^{-126} + q^{-128}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

Vassiliev invariants

V2 and V3: (2, 2)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
8 16 32 \frac{172}{3} \frac{20}{3} 128 \frac{544}{3} \frac{64}{3} 16 \frac{256}{3} 128 \frac{1376}{3} \frac{160}{3} \frac{10471}{15} \frac{1636}{15} \frac{5404}{45} \frac{89}{9} -\frac{89}{15}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=2 is the signature of K11a171. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          4 4
13         81 -7
11        114  7
9       158   -7
7      1511    4
5     1415     1
3    1215      -3
1   715       8
-1  411        -7
-3 17         6
-5 4          -4
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=2 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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K11a170.gif

K11a170

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K11a172