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

Visit K11a65 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X16,5,17,6 X10,8,11,7 X2,9,3,10 X22,11,1,12 X20,13,21,14 X18,15,19,16 X6,17,7,18 X14,19,15,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -11, 7, -10, 8, -3, 9, -8, 10, -7, 11, -6
Dowker-Thistlethwaite code 4 8 16 10 2 22 20 18 6 14 12
A Braid Representative
A Morse Link Presentation K11a65 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a65/ThurstonBennequinNumber
Hyperbolic Volume 10.6865
A-Polynomial See Data:K11a65/A-polynomial

[edit Notes for K11a65's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a65's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^2+15 t-23+15 t^{-1} -3 t^{-2}
Conway polynomial -3 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 59, -2 }
Jones polynomial q-2+4 q^{-1} -6 q^{-2} +8 q^{-3} -9 q^{-4} +9 q^{-5} -7 q^{-6} +6 q^{-7} -4 q^{-8} +2 q^{-9} - q^{-10}
HOMFLY-PT polynomial (db, data sources) -a^{10}+2 z^2 a^8+a^8-z^4 a^6-z^4 a^4+z^2 a^4+a^4-z^4 a^2-z^2 a^2-a^2+z^2+1
Kauffman polynomial (db, data sources) z^7 a^{11}-5 z^5 a^{11}+7 z^3 a^{11}-3 z a^{11}+2 z^8 a^{10}-9 z^6 a^{10}+11 z^4 a^{10}-5 z^2 a^{10}+a^{10}+2 z^9 a^9-7 z^7 a^9+3 z^5 a^9+4 z^3 a^9-2 z a^9+z^{10} a^8-z^8 a^8-7 z^6 a^8+9 z^4 a^8-3 z^2 a^8+a^8+4 z^9 a^7-15 z^7 a^7+17 z^5 a^7-8 z^3 a^7+z a^7+z^{10} a^6-z^8 a^6-2 z^6 a^6+z^4 a^6+z^2 a^6+2 z^9 a^5-5 z^7 a^5+7 z^5 a^5-4 z^3 a^5+2 z^8 a^4-2 z^6 a^4+2 z^4 a^4-2 z^2 a^4+a^4+2 z^7 a^3-2 z^3 a^3+z a^3+2 z^6 a^2-3 z^2 a^2+a^2+2 z^5 a-3 z^3 a+z a+z^4-2 z^2+1
The A2 invariant Data:K11a65/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a65/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: (3, -9)
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
12 -72 72 366 58 -864 -2128 -320 -392 288 2592 4392 696 \frac{127231}{10} -\frac{10786}{15} \frac{91862}{15} \frac{1793}{6} \frac{8351}{10}

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 K11a65. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3           11
1          1 -1
-1         31 2
-3        42  -2
-5       42   2
-7      54    -1
-9     44     0
-11    35      2
-13   34       -1
-15  13        2
-17 13         -2
-19 1          1
-211           -1
Integral Khovanov Homology

(db, data source)

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