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

Visit K11a316 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X16,5,17,6 X20,8,21,7 X22,10,1,9 X4,12,5,11 X18,13,19,14 X10,15,11,16 X2,17,3,18 X14,19,15,20 X8,22,9,21
Gauss code 1, -9, 2, -6, 3, -1, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -7, 10, -4, 11, -5
Dowker-Thistlethwaite code 6 12 16 20 22 4 18 10 2 14 8
A Braid Representative
A Morse Link Presentation K11a316 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:K11a316/ThurstonBennequinNumber
Hyperbolic Volume 15.8582
A-Polynomial See Data:K11a316/A-polynomial

[edit Notes for K11a316'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 K11a316's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+14 t^2-25 t+31-25 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+4 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 121, 0 }
Jones polynomial q^6-4 q^5+8 q^4-13 q^3+17 q^2-19 q+20-16 q^{-1} +12 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +15 z^4-6 a^2 z^2-11 z^2 a^{-2} +2 z^2 a^{-4} +17 z^2-3 a^2-5 a^{-2} + a^{-4} +8
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10}+6 a z^9+11 z^9 a^{-1} +5 z^9 a^{-3} +8 a^2 z^8+7 z^8 a^{-2} +6 z^8 a^{-4} +9 z^8+6 a^3 z^7-10 a z^7-26 z^7 a^{-1} -6 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-21 a^2 z^6-26 z^6 a^{-2} -13 z^6 a^{-4} +z^6 a^{-6} -36 z^6+a^5 z^5-13 a^3 z^5+4 a z^5+23 z^5 a^{-1} -5 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+27 a^2 z^4+27 z^4 a^{-2} +5 z^4 a^{-4} -2 z^4 a^{-6} +52 z^4-2 a^5 z^3+10 a^3 z^3+10 a z^3-5 z^3 a^{-1} +3 z^3 a^{-3} +6 z^3 a^{-5} -14 a^2 z^2-18 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -30 z^2-3 a^3 z-5 a z-3 z a^{-1} -z a^{-3} +3 a^2+5 a^{-2} + a^{-4} +8
The A2 invariant -q^{14}+q^{12}-3 q^{10}+q^8+q^6-2 q^4+6 q^2-1+4 q^{-2} -2 q^{-6} +2 q^{-8} -4 q^{-10} + q^{-12} - q^{-16} + q^{-18}
The G2 invariant Data:K11a316/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a35,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a35, K11a36,}

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{28}{3} -\frac{52}{3} -64 -\frac{464}{3} \frac{256}{3} -168 \frac{256}{3} 32 \frac{224}{3} -\frac{416}{3} \frac{751}{15} \frac{4156}{15} -\frac{15116}{45} -\frac{415}{9} -\frac{1649}{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 K11a316. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13           11
11          3 -3
9         51 4
7        83  -5
5       95   4
3      108    -2
1     109     1
-1    711      4
-3   59       -4
-5  27        5
-7 15         -4
-9 2          2
-111           -1
Integral Khovanov Homology

(db, data source)

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