K11a180

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K11a179.gif

K11a179

K11a181.gif

K11a181

Contents

K11a180.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a180 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X16,8,17,7 X18,9,19,10 X20,11,21,12 X2,13,3,14 X6,16,7,15 X22,18,1,17 X10,19,11,20 X8,21,9,22
Gauss code 1, -7, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code 4 12 14 16 18 20 2 6 22 10 8
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation K11a180 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a180/ThurstonBennequinNumber
Hyperbolic Volume 12.7428
A-Polynomial See Data:K11a180/A-polynomial

[edit Notes for K11a180's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a180's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+11 t^2-17 t+21-17 t^{-1} +11 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 89, 0 }
Jones polynomial -q^5+3 q^4-5 q^3+9 q^2-12 q+14-14 q^{-1} +12 q^{-2} -9 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) z^8-2 a^2 z^6-z^6 a^{-2} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +13 z^4+3 a^4 z^2-12 a^2 z^2-4 z^2 a^{-2} +11 z^2+2 a^4-4 a^2+3
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+3 a^3 z^9+6 a z^9+3 z^9 a^{-1} +4 a^4 z^8+5 a^2 z^8+4 z^8 a^{-2} +5 z^8+3 a^5 z^7-5 a^3 z^7-16 a z^7-4 z^7 a^{-1} +4 z^7 a^{-3} +a^6 z^6-11 a^4 z^6-22 a^2 z^6-7 z^6 a^{-2} +3 z^6 a^{-4} -20 z^6-9 a^5 z^5-a^3 z^5+18 a z^5+z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+8 a^4 z^4+31 a^2 z^4+4 z^4 a^{-2} -7 z^4 a^{-4} +31 z^4+6 a^5 z^3+a^3 z^3-6 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-5 a^4 z^2-20 a^2 z^2+3 z^2 a^{-4} -16 z^2-a^5 z-2 z a^{-1} -z a^{-3} +2 a^4+4 a^2+3
The A2 invariant q^{18}+q^{12}-2 q^{10}+2 q^8-q^6-q^4+q^2-3+3 q^{-2} - q^{-4} +2 q^{-6} +2 q^{-8} - q^{-10} + q^{-12} - q^{-14}
The G2 invariant q^{94}-2 q^{92}+5 q^{90}-9 q^{88}+10 q^{86}-9 q^{84}+16 q^{80}-32 q^{78}+48 q^{76}-53 q^{74}+38 q^{72}-5 q^{70}-43 q^{68}+94 q^{66}-122 q^{64}+121 q^{62}-76 q^{60}-2 q^{58}+89 q^{56}-155 q^{54}+177 q^{52}-139 q^{50}+54 q^{48}+47 q^{46}-126 q^{44}+151 q^{42}-112 q^{40}+33 q^{38}+56 q^{36}-112 q^{34}+105 q^{32}-46 q^{30}-52 q^{28}+139 q^{26}-177 q^{24}+144 q^{22}-49 q^{20}-78 q^{18}+186 q^{16}-241 q^{14}+217 q^{12}-125 q^{10}-9 q^8+131 q^6-206 q^4+206 q^2-135+32 q^{-2} +67 q^{-4} -120 q^{-6} +113 q^{-8} -52 q^{-10} -25 q^{-12} +91 q^{-14} -105 q^{-16} +68 q^{-18} +8 q^{-20} -85 q^{-22} +140 q^{-24} -141 q^{-26} +99 q^{-28} -27 q^{-30} -53 q^{-32} +108 q^{-34} -132 q^{-36} +119 q^{-38} -74 q^{-40} +21 q^{-42} +27 q^{-44} -62 q^{-46} +75 q^{-48} -70 q^{-50} +50 q^{-52} -23 q^{-54} -3 q^{-56} +22 q^{-58} -31 q^{-60} +30 q^{-62} -21 q^{-64} +13 q^{-66} -2 q^{-68} -5 q^{-70} +6 q^{-72} -7 q^{-74} +4 q^{-76} -2 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, 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{68}{3} \frac{28}{3} -128 -\frac{704}{3} -\frac{320}{3} -16 -\frac{256}{3} 128 -\frac{544}{3} -\frac{224}{3} \frac{3089}{15} -\frac{2356}{15} \frac{12356}{45} -\frac{65}{9} \frac{929}{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 K11a180. 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          2 2
7         31 -2
5        62  4
3       63   -3
1      86    2
-1     77     0
-3    57      -2
-5   47       3
-7  25        -3
-9 14         3
-11 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

Read me first: Modifying Knot Pages.

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

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11a179

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K11a181