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

Visit K11a321 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X12,3,13,4 X16,5,17,6 X22,8,1,7 X18,9,19,10 X20,11,21,12 X2,13,3,14 X8,15,9,16 X4,17,5,18 X10,19,11,20 X14,21,15,22
Gauss code 1, -7, 2, -9, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -3, 9, -5, 10, -6, 11, -4
Dowker-Thistlethwaite code 6 12 16 22 18 20 2 8 4 10 14
A Braid Representative
A Morse Link Presentation K11a321 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a321/ThurstonBennequinNumber
Hyperbolic Volume 16.2583
A-Polynomial See Data:K11a321/A-polynomial

[edit Notes for K11a321'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 K11a321's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+15 t^2-27 t+31-27 t^{-1} +15 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-3 z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{11,t+1\}
Determinant and Signature { 121, -4 }
Jones polynomial 1-3 q^{-1} +7 q^{-2} -12 q^{-3} +17 q^{-4} -19 q^{-5} +20 q^{-6} -17 q^{-7} +13 q^{-8} -8 q^{-9} +3 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-2 a^{10}+3 z^4 a^8+7 z^2 a^8+3 a^8-2 z^6 a^6-6 z^4 a^6-5 z^2 a^6-2 a^6-z^6 a^4-z^4 a^4+3 z^2 a^4+2 a^4+z^4 a^2+2 z^2 a^2
Kauffman polynomial (db, data sources) z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+6 z^7 a^{11}-10 z^5 a^{11}+8 z^3 a^{11}-4 z a^{11}+7 z^8 a^{10}-10 z^6 a^{10}+7 z^4 a^{10}-5 z^2 a^{10}+2 a^{10}+5 z^9 a^9-z^7 a^9-10 z^5 a^9+12 z^3 a^9-4 z a^9+2 z^{10} a^8+7 z^8 a^8-18 z^6 a^8+16 z^4 a^8-8 z^2 a^8+3 a^8+10 z^9 a^7-21 z^7 a^7+20 z^5 a^7-11 z^3 a^7+2 z a^7+2 z^{10} a^6+5 z^8 a^6-19 z^6 a^6+20 z^4 a^6-11 z^2 a^6+2 a^6+5 z^9 a^5-11 z^7 a^5+11 z^5 a^5-8 z^3 a^5+z a^5+5 z^8 a^4-13 z^6 a^4+12 z^4 a^4-7 z^2 a^4+2 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^2
The A2 invariant Data:K11a321/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a321/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: (6, -16)
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
24 -128 288 860 164 -3072 -\frac{18656}{3} -\frac{3200}{3} -1120 2304 8192 20640 3936 \frac{225791}{5} -\frac{28612}{15} \frac{331804}{15} \frac{1649}{3} \frac{16111}{5}

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 K11a321. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
1           11
-1          2 -2
-3         51 4
-5        83  -5
-7       94   5
-9      108    -2
-11     109     1
-13    710      3
-15   610       -4
-17  27        5
-19 16         -5
-21 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-3 {\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=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
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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