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

Visit K11a36 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X14,6,15,5 X2837 X16,9,17,10 X20,11,21,12 X6,14,7,13 X12,15,13,16 X22,17,1,18 X10,19,11,20 X18,21,19,22
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -8, 7, -3, 8, -5, 9, -11, 10, -6, 11, -9
Dowker-Thistlethwaite code 4 8 14 2 16 20 6 12 22 10 18
A Braid Representative
A Morse Link Presentation K11a36 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:K11a36/ThurstonBennequinNumber
Hyperbolic Volume 15.1597
A-Polynomial See Data:K11a36/A-polynomial

[edit Notes for K11a36's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a36's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+12 t^2-28 t+37-28 t^{-1} +12 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 121, 0 }
Jones polynomial -q^5+3 q^4-7 q^3+12 q^2-16 q+20-19 q^{-1} +17 q^{-2} -13 q^{-3} +8 q^{-4} -4 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6-z^6+a^4 z^4-2 a^2 z^4+2 z^4 a^{-2} -z^4+a^4 z^2-2 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +z^2-a^2+ a^{-2} - a^{-4} +2
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+4 a^3 z^9+7 a z^9+3 z^9 a^{-1} +6 a^4 z^8+12 a^2 z^8+5 z^8 a^{-2} +11 z^8+4 a^5 z^7+a^3 z^7+8 z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-13 a^4 z^6-28 a^2 z^6-2 z^6 a^{-2} +3 z^6 a^{-4} -19 z^6-10 a^5 z^5-20 a^3 z^5-25 a z^5-23 z^5 a^{-1} -7 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+6 a^4 z^4+15 a^2 z^4-5 z^4 a^{-2} -5 z^4 a^{-4} +7 z^4+7 a^5 z^3+18 a^3 z^3+25 a z^3+20 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +a^6 z^2-3 a^2 z^2+5 z^2 a^{-2} +3 z^2 a^{-4} -a^5 z-5 a^3 z-8 a z-6 z a^{-1} -z a^{-3} +z a^{-5} +a^2- a^{-2} - a^{-4} +2
The A2 invariant Data:K11a36/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a36/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a169,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a35, K11a316,}

Vassiliev invariants

V2 and V3: (2, 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
8 8 32 \frac{172}{3} \frac{44}{3} 64 \frac{272}{3} -\frac{64}{3} 40 \frac{256}{3} 32 \frac{1376}{3} \frac{352}{3} \frac{9271}{15} -\frac{244}{15} \frac{13924}{45} \frac{281}{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 K11a36. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           1-1
9          2 2
7         51 -4
5        72  5
3       95   -4
1      117    4
-1     910     1
-3    810      -2
-5   59       4
-7  38        -5
-9 15         4
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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