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

Visit K11a326 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X18,5,19,6 X14,8,15,7 X22,10,1,9 X4,12,5,11 X20,13,21,14 X8,16,9,15 X10,17,11,18 X2,19,3,20 X16,21,17,22
Gauss code 1, -10, 2, -6, 3, -1, 4, -8, 5, -9, 6, -2, 7, -4, 8, -11, 9, -3, 10, -7, 11, -5
Dowker-Thistlethwaite code 6 12 18 14 22 4 20 8 10 2 16
A Braid Representative
A Morse Link Presentation K11a326 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a326/ThurstonBennequinNumber
Hyperbolic Volume 18.6893
A-Polynomial See Data:K11a326/A-polynomial

[edit Notes for K11a326'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 K11a326's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+19 t^2-36 t+45-36 t^{-1} +19 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+3 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 169, 0 }
Jones polynomial q^6-5 q^5+11 q^4-18 q^3+24 q^2-27 q+28-23 q^{-1} +17 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} +11 z^2-2 a^2-2 a^{-2} +5
Kauffman polynomial (db, data sources) 4 z^{10} a^{-2} +4 z^{10}+11 a z^9+21 z^9 a^{-1} +10 z^9 a^{-3} +13 a^2 z^8+13 z^8 a^{-2} +10 z^8 a^{-4} +16 z^8+9 a^3 z^7-14 a z^7-44 z^7 a^{-1} -16 z^7 a^{-3} +5 z^7 a^{-5} +4 a^4 z^6-24 a^2 z^6-47 z^6 a^{-2} -21 z^6 a^{-4} +z^6 a^{-6} -53 z^6+a^5 z^5-13 a^3 z^5+3 a z^5+28 z^5 a^{-1} +2 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+22 a^2 z^4+39 z^4 a^{-2} +11 z^4 a^{-4} -z^4 a^{-6} +53 z^4-a^5 z^3+7 a^3 z^3+6 a z^3-5 z^3 a^{-1} +3 z^3 a^{-5} -10 a^2 z^2-14 z^2 a^{-2} -2 z^2 a^{-4} -22 z^2-2 a^3 z-3 a z-z a^{-1} +z a^{-3} +z a^{-5} +2 a^2+2 a^{-2} +5
The A2 invariant -q^{14}+2 q^{12}-4 q^{10}+2 q^8+q^6-4 q^4+7 q^2-3+5 q^{-2} -2 q^{-6} +4 q^{-8} -5 q^{-10} +2 q^{-12} -2 q^{-16} + q^{-18}
The G2 invariant Data:K11a326/QuantumInvariant/G2/1,0

"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, 0)
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 0 32 \frac{76}{3} -\frac{28}{3} 0 -32 64 -96 \frac{256}{3} 0 \frac{608}{3} -\frac{224}{3} \frac{2551}{15} \frac{3076}{15} -\frac{8756}{45} -\frac{295}{9} -\frac{1049}{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 K11a326. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13           11
11          4 -4
9         71 6
7        114  -7
5       137   6
3      1411    -3
1     1413     1
-1    1015      5
-3   713       -6
-5  310        7
-7 17         -6
-9 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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