K11a164

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K11a163.gif

K11a163

K11a165.gif

K11a165

Contents

K11a164.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a164 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X18,6,19,5 X14,8,15,7 X16,9,17,10 X2,11,3,12 X20,13,21,14 X22,16,1,15 X12,17,13,18 X6,20,7,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -9, 7, -4, 8, -5, 9, -3, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 18 14 16 2 20 22 12 6 8
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation K11a164 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a164/ThurstonBennequinNumber
Hyperbolic Volume 18.3585
A-Polynomial See Data:K11a164/A-polynomial

[edit Notes for K11a164's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a164's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-7 t^3+20 t^2-35 t+43-35 t^{-1} +20 t^{-2} -7 t^{-3} + t^{-4}
Conway polynomial z^8+z^6-2 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 169, 0 }
Jones polynomial -q^5+5 q^4-11 q^3+18 q^2-24 q+28-27 q^{-1} +23 q^{-2} -17 q^{-3} +10 q^{-4} -4 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) z^8-2 a^2 z^6-z^6 a^{-2} +4 z^6+a^4 z^4-6 a^2 z^4-2 z^4 a^{-2} +5 z^4+2 a^4 z^2-5 a^2 z^2+z^2+a^4-a^2+ a^{-2}
Kauffman polynomial (db, data sources) 3 a^2 z^{10}+3 z^{10}+8 a^3 z^9+18 a z^9+10 z^9 a^{-1} +8 a^4 z^8+16 a^2 z^8+14 z^8 a^{-2} +22 z^8+4 a^5 z^7-10 a^3 z^7-27 a z^7-2 z^7 a^{-1} +11 z^7 a^{-3} +a^6 z^6-17 a^4 z^6-49 a^2 z^6-19 z^6 a^{-2} +5 z^6 a^{-4} -55 z^6-8 a^5 z^5-3 a^3 z^5+a z^5-19 z^5 a^{-1} -14 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+13 a^4 z^4+41 a^2 z^4+6 z^4 a^{-2} -4 z^4 a^{-4} +36 z^4+5 a^5 z^3+5 a^3 z^3+8 a z^3+12 z^3 a^{-1} +4 z^3 a^{-3} +a^6 z^2-6 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -5 z^2-a^5 z-a^3 z-a z-z a^{-1} +a^4+a^2- a^{-2}
The A2 invariant q^{18}-q^{16}+3 q^{12}-4 q^{10}+4 q^8-2 q^6-2 q^4+4 q^2-5+6 q^{-2} -4 q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} +3 q^{-12} - q^{-14}
The G2 invariant q^{94}-3 q^{92}+8 q^{90}-16 q^{88}+23 q^{86}-28 q^{84}+20 q^{82}+10 q^{80}-63 q^{78}+139 q^{76}-210 q^{74}+232 q^{72}-166 q^{70}-19 q^{68}+310 q^{66}-612 q^{64}+808 q^{62}-756 q^{60}+371 q^{58}+267 q^{56}-968 q^{54}+1451 q^{52}-1462 q^{50}+947 q^{48}-31 q^{46}-947 q^{44}+1580 q^{42}-1599 q^{40}+975 q^{38}+18 q^{36}-939 q^{34}+1370 q^{32}-1120 q^{30}+316 q^{28}+702 q^{26}-1454 q^{24}+1593 q^{22}-1032 q^{20}-64 q^{18}+1251 q^{16}-2080 q^{14}+2211 q^{12}-1558 q^{10}+365 q^8+973 q^6-1971 q^4+2260 q^2-1767+683 q^{-2} +538 q^{-4} -1418 q^{-6} +1613 q^{-8} -1076 q^{-10} +123 q^{-12} +820 q^{-14} -1311 q^{-16} +1130 q^{-18} -397 q^{-20} -574 q^{-22} +1335 q^{-24} -1563 q^{-26} +1201 q^{-28} -394 q^{-30} -496 q^{-32} +1148 q^{-34} -1369 q^{-36} +1146 q^{-38} -627 q^{-40} +27 q^{-42} +446 q^{-44} -687 q^{-46} +681 q^{-48} -496 q^{-50} +252 q^{-52} -18 q^{-54} -141 q^{-56} +203 q^{-58} -199 q^{-60} +141 q^{-62} -73 q^{-64} +21 q^{-66} +16 q^{-68} -28 q^{-70} +27 q^{-72} -20 q^{-74} +10 q^{-76} -4 q^{-78} + q^{-80}

"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, 1)
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 8 32 \frac{164}{3} \frac{100}{3} -64 -\frac{400}{3} -\frac{160}{3} -24 -\frac{256}{3} 32 -\frac{1312}{3} -\frac{800}{3} -\frac{4111}{15} \frac{908}{5} -\frac{21124}{45} \frac{655}{9} -\frac{1951}{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=0 is the signature of K11a164. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
11           1-1
9          4 4
7         71 -6
5        114  7
3       137   -6
1      1511    4
-1     1314     1
-3    1014      -4
-5   713       6
-7  310        -7
-9 17         6
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{15}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\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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K11a163.gif

K11a163

K11a165.gif

K11a165