K11a76

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K11a75.gif

K11a75

K11a77.gif

K11a77

Contents

K11a76.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a76 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X18,9,19,10 X2,11,3,12 X20,14,21,13 X8,15,9,16 X22,18,1,17 X6,19,7,20 X16,22,17,21
Gauss code 1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -4, 8, -11, 9, -5, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 10 12 14 18 2 20 8 22 6 16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a76 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:K11a76/ThurstonBennequinNumber
Hyperbolic Volume 16.9805
A-Polynomial See Data:K11a76/A-polynomial

[edit Notes for K11a76'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 K11a76's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+17 t^2-30 t+37-30 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 145, 0 }
Jones polynomial -q^5+4 q^4-9 q^3+15 q^2-20 q+24-23 q^{-1} +20 q^{-2} -15 q^{-3} +9 q^{-4} -4 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-7 a^2 z^4-3 z^4 a^{-2} +10 z^4+2 a^4 z^2-8 a^2 z^2-3 z^2 a^{-2} +9 z^2+a^4-3 a^2- a^{-2} +4
Kauffman polynomial (db, data sources) 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+13 a z^9+7 z^9 a^{-1} +7 a^4 z^8+14 a^2 z^8+10 z^8 a^{-2} +17 z^8+4 a^5 z^7-5 a^3 z^7-19 a z^7-2 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-15 a^4 z^6-41 a^2 z^6-16 z^6 a^{-2} +4 z^6 a^{-4} -45 z^6-9 a^5 z^5-10 a^3 z^5+a z^5-11 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+9 a^4 z^4+35 a^2 z^4+12 z^4 a^{-2} -5 z^4 a^{-4} +41 z^4+6 a^5 z^3+9 a^3 z^3+6 a z^3+10 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-3 a^4 z^2-15 a^2 z^2-5 z^2 a^{-2} +z^2 a^{-4} -17 z^2-a^5 z-2 a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^4+3 a^2+ a^{-2} +4
The A2 invariant q^{18}-q^{16}+2 q^{12}-4 q^{10}+3 q^8-2 q^6-q^4+4 q^2-3+6 q^{-2} -3 q^{-4} + q^{-6} +2 q^{-8} -3 q^{-10} +2 q^{-12} - q^{-14}
The G2 invariant q^{94}-3 q^{92}+8 q^{90}-16 q^{88}+22 q^{86}-25 q^{84}+14 q^{82}+18 q^{80}-67 q^{78}+130 q^{76}-176 q^{74}+169 q^{72}-88 q^{70}-77 q^{68}+294 q^{66}-480 q^{64}+556 q^{62}-446 q^{60}+129 q^{58}+307 q^{56}-717 q^{54}+935 q^{52}-836 q^{50}+440 q^{48}+134 q^{46}-670 q^{44}+941 q^{42}-842 q^{40}+406 q^{38}+170 q^{36}-633 q^{34}+764 q^{32}-509 q^{30}-20 q^{28}+597 q^{26}-952 q^{24}+914 q^{22}-475 q^{20}-231 q^{18}+920 q^{16}-1333 q^{14}+1302 q^{12}-812 q^{10}+53 q^8+721 q^6-1226 q^4+1288 q^2-905+241 q^{-2} +429 q^{-4} -839 q^{-6} +843 q^{-8} -458 q^{-10} -93 q^{-12} +573 q^{-14} -751 q^{-16} +553 q^{-18} -85 q^{-20} -462 q^{-22} +844 q^{-24} -903 q^{-26} +640 q^{-28} -155 q^{-30} -349 q^{-32} +694 q^{-34} -793 q^{-36} +646 q^{-38} -344 q^{-40} +3 q^{-42} +259 q^{-44} -394 q^{-46} +389 q^{-48} -285 q^{-50} +149 q^{-52} -15 q^{-54} -76 q^{-56} +114 q^{-58} -115 q^{-60} +84 q^{-62} -46 q^{-64} +16 q^{-66} +7 q^{-68} -16 q^{-70} +16 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a160, K11a289,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a160, K11a289,}

Vassiliev invariants

V2 and V3: (0, 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
0 8 0 -16 -8 0 -\frac{16}{3} -\frac{160}{3} 40 0 32 0 0 72 -24 \frac{344}{3} -\frac{136}{3} 24

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 K11a76. 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          3 3
7         61 -5
5        93  6
3       116   -5
1      139    4
-1     1112     1
-3    912      -3
-5   611       5
-7  39        -6
-9 16         5
-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}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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).

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K11a75.gif

K11a75

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K11a77