K11a71

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K11a70.gif

K11a70

K11a72.gif

K11a72

Contents

K11a71.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a71 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X12,5,13,6 X14,8,15,7 X2,10,3,9 X22,11,1,12 X18,13,19,14 X20,16,21,15 X6,18,7,17 X8,19,9,20 X16,22,17,21
Gauss code 1, -5, 2, -1, 3, -9, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -6
Dowker-Thistlethwaite code 4 10 12 14 2 22 18 20 6 8 16
A Braid Representative
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BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation K11a71 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a71/ThurstonBennequinNumber
Hyperbolic Volume 17.3873
A-Polynomial See Data:K11a71/A-polynomial

[edit Notes for K11a71'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 K11a71's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-18 t^2+34 t-41+34 t^{-1} -18 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6-2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 159, 2 }
Jones polynomial -q^8+5 q^7-11 q^6+17 q^5-23 q^4+26 q^3-25 q^2+22 q-15+9 q^{-1} -4 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-10 z^4 a^{-2} +6 z^4 a^{-4} -z^4 a^{-6} +3 z^4-7 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} +3 z^2+1
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10} a^{-4} +6 z^9 a^{-1} +14 z^9 a^{-3} +8 z^9 a^{-5} +17 z^8 a^{-2} +23 z^8 a^{-4} +13 z^8 a^{-6} +7 z^8+4 a z^7-3 z^7 a^{-1} -13 z^7 a^{-3} +5 z^7 a^{-5} +11 z^7 a^{-7} +a^2 z^6-47 z^6 a^{-2} -53 z^6 a^{-4} -17 z^6 a^{-6} +5 z^6 a^{-8} -15 z^6-9 a z^5-14 z^5 a^{-1} -18 z^5 a^{-3} -29 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+39 z^4 a^{-2} +35 z^4 a^{-4} +5 z^4 a^{-6} -4 z^4 a^{-8} +11 z^4+6 a z^3+14 z^3 a^{-1} +20 z^3 a^{-3} +17 z^3 a^{-5} +5 z^3 a^{-7} +a^2 z^2-12 z^2 a^{-2} -8 z^2 a^{-4} -z^2 a^{-6} -4 z^2-a z-3 z a^{-1} -3 z a^{-3} -z a^{-5} +1
The A2 invariant q^8-2 q^6+3 q^4-2 q^2-1+5 q^{-2} -4 q^{-4} +6 q^{-6} -2 q^{-8} + q^{-12} -5 q^{-14} +4 q^{-16} -2 q^{-18} +2 q^{-22} - q^{-24}
The G2 invariant q^{46}-3 q^{44}+8 q^{42}-16 q^{40}+22 q^{38}-25 q^{36}+14 q^{34}+18 q^{32}-65 q^{30}+127 q^{28}-176 q^{26}+178 q^{24}-110 q^{22}-51 q^{20}+277 q^{18}-495 q^{16}+619 q^{14}-551 q^{12}+254 q^{10}+221 q^8-733 q^6+1094 q^4-1121 q^2+759-102 q^{-2} -637 q^{-4} +1158 q^{-6} -1252 q^{-8} +878 q^{-10} -176 q^{-12} -542 q^{-14} +971 q^{-16} -920 q^{-18} +406 q^{-20} +341 q^{-22} -974 q^{-24} +1199 q^{-26} -873 q^{-28} +94 q^{-30} +840 q^{-32} -1553 q^{-34} +1765 q^{-36} -1353 q^{-38} +450 q^{-40} +627 q^{-42} -1496 q^{-44} +1839 q^{-46} -1553 q^{-48} +761 q^{-50} +212 q^{-52} -989 q^{-54} +1279 q^{-56} -1014 q^{-58} +349 q^{-60} +402 q^{-62} -898 q^{-64} +913 q^{-66} -467 q^{-68} -244 q^{-70} +900 q^{-72} -1206 q^{-74} +1053 q^{-76} -495 q^{-78} -230 q^{-80} +847 q^{-82} -1157 q^{-84} +1079 q^{-86} -685 q^{-88} +156 q^{-90} +320 q^{-92} -610 q^{-94} +664 q^{-96} -522 q^{-98} +289 q^{-100} -49 q^{-102} -126 q^{-104} +202 q^{-106} -205 q^{-108} +147 q^{-110} -76 q^{-112} +22 q^{-114} +16 q^{-116} -28 q^{-118} +27 q^{-120} -20 q^{-122} +10 q^{-124} -4 q^{-126} + q^{-128}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a248,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a248,}

Vassiliev invariants

V2 and V3: (0, 0)
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
0 0 0 16 16 0 32 0 32 0 0 0 0 40 \frac{176}{3} -\frac{160}{3} \frac{104}{3} -40

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 K11a71. 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         71 -6
11        104  6
9       137   -6
7      1310    3
5     1213     1
3    1013      -3
1   613       7
-1  39        -6
-3 16         5
-5 3          -3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

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K11a70.gif

K11a70

K11a72.gif

K11a72