K11a63

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K11a62.gif

K11a62

K11a64.gif

K11a64

Contents

K11a63.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a63 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X16,5,17,6 X10,8,11,7 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,15,1,16 X6,17,7,18 X14,19,15,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -11, 7, -10, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 16 10 2 18 20 22 6 14 12
A Braid Representative
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A Morse Link Presentation K11a63 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a63/ThurstonBennequinNumber
Hyperbolic Volume 14.0116
A-Polynomial See Data:K11a63/A-polynomial

[edit Notes for K11a63's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 4

[edit Notes for K11a63's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-21 t+25-21 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 93, -4 }
Jones polynomial 1-3 q^{-1} +7 q^{-2} -10 q^{-3} +13 q^{-4} -15 q^{-5} +15 q^{-6} -12 q^{-7} +9 q^{-8} -5 q^{-9} +2 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-2 a^{10}+2 z^4 a^8+5 z^2 a^8+3 a^8-z^6 a^6-2 z^4 a^6-z^2 a^6-a^6-z^6 a^4-2 z^4 a^4+z^4 a^2+2 z^2 a^2+a^2
Kauffman polynomial (db, data sources) z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+3 z^7 a^{11}-5 z^5 a^{11}+2 z^3 a^{11}-z a^{11}+4 z^8 a^{10}-10 z^6 a^{10}+16 z^4 a^{10}-12 z^2 a^{10}+2 a^{10}+3 z^9 a^9-4 z^7 a^9+z^5 a^9+7 z^3 a^9-3 z a^9+z^{10} a^8+6 z^8 a^8-23 z^6 a^8+35 z^4 a^8-18 z^2 a^8+3 a^8+6 z^9 a^7-12 z^7 a^7+6 z^5 a^7+4 z^3 a^7-z a^7+z^{10} a^6+6 z^8 a^6-20 z^6 a^6+18 z^4 a^6-7 z^2 a^6+a^6+3 z^9 a^5-2 z^7 a^5-9 z^5 a^5+7 z^3 a^5-2 z a^5+4 z^8 a^4-8 z^6 a^4+z^2 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 a^3-z a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2
The A2 invariant Data:K11a63/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a63/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a309,}

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

Vassiliev invariants

V2 and V3: (5, -14)
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
20 -112 200 \frac{2038}{3} \frac{314}{3} -2240 -\frac{13792}{3} -\frac{2368}{3} -656 \frac{4000}{3} 6272 \frac{40760}{3} \frac{6280}{3} \frac{191023}{6} \frac{1154}{3} \frac{116558}{9} \frac{5749}{18} \frac{10063}{6}

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=-4 is the signature of K11a63. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1          2 -2
-3         51 4
-5        63  -3
-7       74   3
-9      86    -2
-11     77     0
-13    58      3
-15   47       -3
-17  15        4
-19 14         -3
-21 1          1
-231           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\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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K11a62

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K11a64