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

Visit K11a135 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,6,17,5 X14,7,15,8 X12,10,13,9 X2,11,3,12 X18,14,19,13 X20,15,21,16 X22,18,1,17 X8,20,9,19 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -10, 5, -2, 6, -5, 7, -4, 8, -3, 9, -7, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 16 14 12 2 18 20 22 8 6
A Braid Representative
A Morse Link Presentation K11a135 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a135/ThurstonBennequinNumber
Hyperbolic Volume 17.6564
A-Polynomial See Data:K11a135/A-polynomial

[edit Notes for K11a135's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,3]
Rasmussen s-Invariant 0

[edit Notes for K11a135's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+13 t^2-36 t+51-36 t^{-1} +13 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6+z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 153, 0 }
Jones polynomial q^6-5 q^5+11 q^4-16 q^3+22 q^2-25 q+24-21 q^{-1} +15 q^{-2} -8 q^{-3} +4 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6+2 a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -z^4-a^4 z^2+2 a^2 z^2-3 z^2+2 a^2+2 a^{-2} -3
Kauffman polynomial (db, data sources) 3 z^{10} a^{-2} +3 z^{10}+7 a z^9+16 z^9 a^{-1} +9 z^9 a^{-3} +8 a^2 z^8+13 z^8 a^{-2} +10 z^8 a^{-4} +11 z^8+7 a^3 z^7-2 a z^7-29 z^7 a^{-1} -15 z^7 a^{-3} +5 z^7 a^{-5} +4 a^4 z^6-5 a^2 z^6-41 z^6 a^{-2} -23 z^6 a^{-4} +z^6 a^{-6} -26 z^6+a^5 z^5-9 a^3 z^5-10 a z^5+10 z^5 a^{-1} +z^5 a^{-3} -9 z^5 a^{-5} -6 a^4 z^4-8 a^2 z^4+24 z^4 a^{-2} +12 z^4 a^{-4} -z^4 a^{-6} +9 z^4-a^5 z^3+4 a^3 z^3+11 a z^3+7 z^3 a^{-1} +3 z^3 a^{-3} +2 z^3 a^{-5} +3 a^4 z^2+9 a^2 z^2+3 z^2 a^{-2} +9 z^2-a^3 z-5 a z-5 z a^{-1} -z a^{-3} -2 a^2-2 a^{-2} -3
The A2 invariant Data:K11a135/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a135/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{116}{3} \frac{28}{3} 0 0 32 -32 -\frac{256}{3} 0 -\frac{928}{3} -\frac{224}{3} -\frac{4471}{15} \frac{364}{15} -\frac{9964}{45} \frac{439}{9} -\frac{631}{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 K11a135. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13           11
11          4 -4
9         71 6
7        94  -5
5       137   6
3      129    -3
1     1213     -1
-1    1013      3
-3   511       -6
-5  310        7
-7 15         -4
-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}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
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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