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

Visit K11a93 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X16,7,17,8 X22,10,1,9 X2,11,3,12 X20,13,21,14 X18,15,19,16 X8,17,9,18 X14,19,15,20 X6,22,7,21
Gauss code 1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -8, 10, -7, 11, -5
Dowker-Thistlethwaite code 4 10 12 16 22 2 20 18 8 14 6
A Braid Representative
A Morse Link Presentation K11a93 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a93/ThurstonBennequinNumber
Hyperbolic Volume 13.1112
A-Polynomial See Data:K11a93/A-polynomial

[edit Notes for K11a93'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 K11a93's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+10 t^2-21 t+27-21 t^{-1} +10 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 93, 0 }
Jones polynomial q^4-3 q^3+6 q^2-10 q+13-14 q^{-1} +15 q^{-2} -12 q^{-3} +9 q^{-4} -6 q^{-5} +3 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6-a^6+2 z^4 a^4+4 z^2 a^4+a^4-z^6 a^2-2 z^4 a^2+2 a^2-z^6-3 z^4-4 z^2-2+z^4 a^{-2} +2 z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+6 a^3 z^9+3 a z^9+3 a^6 z^8+5 a^4 z^8+7 a^2 z^8+5 z^8+a^7 z^7-8 a^5 z^7-12 a^3 z^7+3 a z^7+6 z^7 a^{-1} -12 a^6 z^6-25 a^4 z^6-20 a^2 z^6+5 z^6 a^{-2} -2 z^6-4 a^7 z^5-2 a^3 z^5-15 a z^5-6 z^5 a^{-1} +3 z^5 a^{-3} +14 a^6 z^4+26 a^4 z^4+11 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} -7 z^4+5 a^7 z^3+9 a^5 z^3+8 a^3 z^3+9 a z^3+2 z^3 a^{-1} -3 z^3 a^{-3} -5 a^6 z^2-9 a^4 z^2-a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +7 z^2-2 a^7 z-3 a^5 z-2 a^3 z-a z+z a^{-1} +z a^{-3} +a^6+a^4-2 a^2- a^{-2} -2
The A2 invariant Data:K11a93/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a93/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_114,}

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

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{206}{3} \frac{58}{3} -96 -208 0 -56 \frac{32}{3} 288 \frac{824}{3} \frac{232}{3} \frac{28831}{30} -\frac{34}{5} \frac{17942}{45} \frac{641}{18} \frac{991}{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 K11a93. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9           11
7          2 -2
5         41 3
3        62  -4
1       74   3
-1      87    -1
-3     76     1
-5    58      3
-7   47       -3
-9  25        3
-11 14         -3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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