K11a109

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K11a108.gif

K11a108

K11a110.gif

K11a110

Contents

K11a109.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a109 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X16,8,17,7 X2,10,3,9 X20,11,21,12 X22,13,1,14 X18,16,19,15 X8,18,9,17 X6,19,7,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -11, 7, -3, 8, -4, 9, -8, 10, -6, 11, -7
Dowker-Thistlethwaite code 4 10 14 16 2 20 22 18 8 6 12
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11a109 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:K11a109/ThurstonBennequinNumber
Hyperbolic Volume 15.1191
A-Polynomial See Data:K11a109/A-polynomial

[edit Notes for K11a109's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,4]
Rasmussen s-Invariant 0

[edit Notes for K11a109's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+14 t^2-24 t+29-24 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+4 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 117, 0 }
Jones polynomial q^6-4 q^5+7 q^4-12 q^3+17 q^2-18 q+19-16 q^{-1} +12 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +15 z^4-6 a^2 z^2-10 z^2 a^{-2} +2 z^2 a^{-4} +17 z^2-3 a^2-3 a^{-2} +7
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10}+4 a z^9+8 z^9 a^{-1} +4 z^9 a^{-3} +6 a^2 z^8+13 z^8 a^{-2} +6 z^8 a^{-4} +13 z^8+5 a^3 z^7+a z^7-9 z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-9 a^2 z^6-42 z^6 a^{-2} -16 z^6 a^{-4} +z^6 a^{-6} -37 z^6+a^5 z^5-7 a^3 z^5-11 a z^5-10 z^5 a^{-1} -18 z^5 a^{-3} -11 z^5 a^{-5} -5 a^4 z^4+9 a^2 z^4+46 z^4 a^{-2} +13 z^4 a^{-4} -2 z^4 a^{-6} +45 z^4-2 a^5 z^3+3 a^3 z^3+13 a z^3+20 z^3 a^{-1} +19 z^3 a^{-3} +7 z^3 a^{-5} +2 a^4 z^2-8 a^2 z^2-22 z^2 a^{-2} -5 z^2 a^{-4} -27 z^2+a^5 z-a^3 z-6 a z-8 z a^{-1} -4 z a^{-3} +3 a^2+3 a^{-2} +7
The A2 invariant -q^{14}+q^{12}-3 q^{10}+q^8+q^6-2 q^4+5 q^2-2+4 q^{-2} + q^{-4} +3 q^{-8} -4 q^{-10} - q^{-14} - q^{-16} + q^{-18}
The G2 invariant q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+9 q^{72}-8 q^{70}+q^{68}+13 q^{66}-29 q^{64}+47 q^{62}-59 q^{60}+53 q^{58}-30 q^{56}-20 q^{54}+87 q^{52}-151 q^{50}+190 q^{48}-184 q^{46}+111 q^{44}+17 q^{42}-178 q^{40}+321 q^{38}-383 q^{36}+329 q^{34}-167 q^{32}-75 q^{30}+296 q^{28}-427 q^{26}+416 q^{24}-248 q^{22}-q^{20}+233 q^{18}-340 q^{16}+281 q^{14}-81 q^{12}-170 q^{10}+352 q^8-375 q^6+215 q^4+84 q^2-392+602 q^{-2} -592 q^{-4} +372 q^{-6} -11 q^{-8} -369 q^{-10} +626 q^{-12} -669 q^{-14} +501 q^{-16} -172 q^{-18} -178 q^{-20} +435 q^{-22} -489 q^{-24} +346 q^{-26} -84 q^{-28} -187 q^{-30} +334 q^{-32} -310 q^{-34} +123 q^{-36} +138 q^{-38} -351 q^{-40} +438 q^{-42} -346 q^{-44} +100 q^{-46} +172 q^{-48} -392 q^{-50} +469 q^{-52} -393 q^{-54} +203 q^{-56} +24 q^{-58} -208 q^{-60} +303 q^{-62} -291 q^{-64} +199 q^{-66} -76 q^{-68} -33 q^{-70} +95 q^{-72} -115 q^{-74} +96 q^{-76} -55 q^{-78} +22 q^{-80} +7 q^{-82} -18 q^{-84} +17 q^{-86} -14 q^{-88} +7 q^{-90} -3 q^{-92} + q^{-94}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a44, K11a47,}

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

Vassiliev invariants

V2 and V3: (3, 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
12 0 72 78 2 0 32 32 0 288 0 936 24 \frac{9151}{10} \frac{1894}{15} \frac{794}{5} \frac{11}{2} \frac{31}{10}

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=0 is the signature of K11a109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         41 3
7        83  -5
5       94   5
3      98    -1
1     109     1
-1    710      3
-3   59       -4
-5  27        5
-7 15         -4
-9 2          2
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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K11a108.gif

K11a108

K11a110.gif

K11a110