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

Visit K11a317 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X12,3,13,4 X16,5,17,6 X20,8,21,7 X22,10,1,9 X18,11,19,12 X2,13,3,14 X10,15,11,16 X4,17,5,18 X14,19,15,20 X8,22,9,21
Gauss code 1, -7, 2, -9, 3, -1, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -6, 10, -4, 11, -5
Dowker-Thistlethwaite code 6 12 16 20 22 18 2 10 4 14 8
A Braid Representative
A Morse Link Presentation K11a317 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a317/ThurstonBennequinNumber
Hyperbolic Volume 16.0606
A-Polynomial See Data:K11a317/A-polynomial

[edit Notes for K11a317's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a317's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+14 t^2-28 t+35-28 t^{-1} +14 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-4 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{5,t+1\}
Determinant and Signature { 125, 0 }
Jones polynomial q^4-4 q^3+9 q^2-14 q+18-20 q^{-1} +20 q^{-2} -16 q^{-3} +12 q^{-4} -7 q^{-5} +3 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6-2 a^6+3 z^4 a^4+8 z^2 a^4+5 a^4-2 z^6 a^2-7 z^4 a^2-9 z^2 a^2-4 a^2-z^6-z^4+2 z^2+2+z^4 a^{-2} +z^2 a^{-2}
Kauffman polynomial (db, data sources) 2 a^4 z^{10}+2 a^2 z^{10}+4 a^5 z^9+11 a^3 z^9+7 a z^9+3 a^6 z^8+4 a^4 z^8+13 a^2 z^8+12 z^8+a^7 z^7-11 a^5 z^7-26 a^3 z^7-a z^7+13 z^7 a^{-1} -11 a^6 z^6-28 a^4 z^6-42 a^2 z^6+9 z^6 a^{-2} -16 z^6-4 a^7 z^5+5 a^5 z^5+13 a^3 z^5-18 a z^5-18 z^5 a^{-1} +4 z^5 a^{-3} +13 a^6 z^4+35 a^4 z^4+36 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +5 z^4+5 a^7 z^3+5 a^5 z^3-a^3 z^3+9 a z^3+9 z^3 a^{-1} -z^3 a^{-3} -6 a^6 z^2-18 a^4 z^2-18 a^2 z^2+3 z^2 a^{-2} -3 z^2-2 a^7 z-2 a^5 z-a z-z a^{-1} +2 a^6+5 a^4+4 a^2+2
The A2 invariant Data:K11a317/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a317/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a31,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a67,}

Vassiliev invariants

V2 and V3: (1, -3)
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
4 -24 8 \frac{302}{3} \frac{106}{3} -96 -336 0 -152 \frac{32}{3} 288 \frac{1208}{3} \frac{424}{3} \frac{40831}{30} -\frac{1954}{5} \frac{43622}{45} \frac{2561}{18} \frac{4831}{30}

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 K11a317. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9           11
7          3 -3
5         61 5
3        83  -5
1       106   4
-1      119    -2
-3     99     0
-5    711      4
-7   59       -4
-9  27        5
-11 15         -4
-13 2          2
-151           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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