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

Visit K11a193 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X18,7,19,8 X20,9,21,10 X16,11,17,12 X2,13,3,14 X22,16,1,15 X10,17,11,18 X8,19,9,20 X6,21,7,22
Gauss code 1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -3, 8, -6, 9, -4, 10, -5, 11, -8
Dowker-Thistlethwaite code 4 12 14 18 20 16 2 22 10 8 6
A Braid Representative
A Morse Link Presentation K11a193 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:K11a193/ThurstonBennequinNumber
Hyperbolic Volume 13.3532
A-Polynomial See Data:K11a193/A-polynomial

[edit Notes for K11a193's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for K11a193's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-10 t^2+22 t-27+22 t^{-1} -10 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 95, -2 }
Jones polynomial -q^2+3 q-5+10 q^{-1} -13 q^{-2} +15 q^{-3} -15 q^{-4} +13 q^{-5} -10 q^{-6} +6 q^{-7} -3 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-2 z^4 a^6-4 z^2 a^6-2 a^6+z^6 a^4+2 z^4 a^4+z^2 a^4+z^6 a^2+3 z^4 a^2+4 z^2 a^2+2 a^2-z^4-2 z^2
Kauffman polynomial (db, data sources) z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-9 z^5 a^9+7 z^3 a^9-z a^9+4 z^8 a^8-10 z^6 a^8+6 z^4 a^8-2 z^2 a^8+a^8+3 z^9 a^7-4 z^7 a^7-z^5 a^7-z^3 a^7+z a^7+z^{10} a^6+5 z^8 a^6-16 z^6 a^6+17 z^4 a^6-11 z^2 a^6+2 a^6+6 z^9 a^5-14 z^7 a^5+20 z^5 a^5-17 z^3 a^5+5 z a^5+z^{10} a^4+5 z^8 a^4-13 z^6 a^4+15 z^4 a^4-6 z^2 a^4+3 z^9 a^3-3 z^7 a^3+4 z^5 a^3-4 z^3 a^3+3 z a^3+4 z^8 a^2-5 z^6 a^2+5 z^2 a^2-2 a^2+4 z^7 a-7 z^5 a+3 z^3 a+3 z^6-7 z^4+4 z^2+z^5 a^{-1} -2 z^3 a^{-1}
The A2 invariant Data:K11a193/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a193/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: (0, 2)
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 16 0 -80 -16 0 \frac{640}{3} -\frac{128}{3} 112 0 128 0 0 -136 432 -\frac{992}{3} -\frac{392}{3} -56

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=-2 is the signature of K11a193. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5           1-1
3          2 2
1         31 -2
-1        72  5
-3       74   -3
-5      86    2
-7     77     0
-9    68      -2
-11   47       3
-13  26        -4
-15 14         3
-17 2          -2
-191           1
Integral Khovanov Homology

(db, data source)

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