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

Visit K11a11 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X16,8,17,7 X2,9,3,10 X20,11,21,12 X18,14,19,13 X6,16,7,15 X14,18,15,17 X22,19,1,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -11, 7, -9, 8, -4, 9, -7, 10, -6, 11, -10
Dowker-Thistlethwaite code 4 8 10 16 2 20 18 6 14 22 12
A Braid Representative
A Morse Link Presentation K11a11 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a11/ThurstonBennequinNumber
Hyperbolic Volume 14.7239
A-Polynomial See Data:K11a11/A-polynomial

[edit Notes for K11a11'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 K11a11's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-26 t+35-26 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 113, 0 }
Jones polynomial q^6-4 q^5+8 q^4-12 q^3+16 q^2-18 q+18-15 q^{-1} +11 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6+2 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -2 z^4-a^4 z^2+4 a^2 z^2-z^2 a^{-2} +z^2 a^{-4} -3 z^2-a^4+3 a^2+ a^{-2} -2
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10}+3 a z^9+7 z^9 a^{-1} +4 z^9 a^{-3} +4 a^2 z^8+11 z^8 a^{-2} +6 z^8 a^{-4} +9 z^8+4 a^3 z^7+3 a z^7-6 z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-30 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -18 z^6+a^5 z^5-4 a^3 z^5-10 a z^5-11 z^5 a^{-1} -16 z^5 a^{-3} -10 z^5 a^{-5} -6 a^4 z^4-9 a^2 z^4+21 z^4 a^{-2} +7 z^4 a^{-4} -2 z^4 a^{-6} +9 z^4-2 a^5 z^3-a^3 z^3+8 a z^3+15 z^3 a^{-1} +14 z^3 a^{-3} +6 z^3 a^{-5} +4 a^4 z^2+9 a^2 z^2-3 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} +4 z^2+a^5 z+a^3 z-2 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} -a^4-3 a^2- a^{-2} -2
The A2 invariant Data:K11a11/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a11/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a167,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a167,}

Vassiliev invariants

V2 and V3: (0, -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
0 -8 0 16 8 0 \frac{112}{3} \frac{64}{3} 24 0 32 0 0 88 104 -\frac{248}{3} \frac{88}{3} -40

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 K11a11. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13           11
11          3 -3
9         51 4
7        73  -4
5       95   4
3      97    -2
1     99     0
-1    710      3
-3   48       -4
-5  27        5
-7 14         -3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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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