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

Visit K11a261 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X8394 X14,5,15,6 X16,8,17,7 X4,9,5,10 X18,11,19,12 X20,13,21,14 X2,15,3,16 X22,18,1,17 X12,19,13,20 X10,21,11,22
Gauss code 1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -4, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code 6 8 14 16 4 18 20 2 22 12 10
A Braid Representative
A Morse Link Presentation K11a261 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a261/ThurstonBennequinNumber
Hyperbolic Volume 15.9423
A-Polynomial See Data:K11a261/A-polynomial

[edit Notes for K11a261's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a261's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+16 t^2-26 t+31-26 t^{-1} +16 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 129, -4 }
Jones polynomial -q+4-7 q^{-1} +13 q^{-2} -17 q^{-3} +20 q^{-4} -21 q^{-5} +18 q^{-6} -14 q^{-7} +9 q^{-8} -4 q^{-9} + q^{-10}
HOMFLY-PT polynomial (db, data sources) z^4 a^8+2 z^2 a^8+a^8-2 z^6 a^6-7 z^4 a^6-7 z^2 a^6-2 a^6+z^8 a^4+5 z^6 a^4+9 z^4 a^4+6 z^2 a^4-z^6 a^2-3 z^4 a^2-z^2 a^2+2 a^2
Kauffman polynomial (db, data sources) z^4 a^{12}+4 z^5 a^{11}-z^3 a^{11}+9 z^6 a^{10}-8 z^4 a^{10}+3 z^2 a^{10}+13 z^7 a^9-18 z^5 a^9+10 z^3 a^9-2 z a^9+12 z^8 a^8-15 z^6 a^8+3 z^4 a^8-2 z^2 a^8+a^8+7 z^9 a^7+z^7 a^7-24 z^5 a^7+14 z^3 a^7-2 z a^7+2 z^{10} a^6+15 z^8 a^6-48 z^6 a^6+39 z^4 a^6-15 z^2 a^6+2 a^6+12 z^9 a^5-28 z^7 a^5+12 z^5 a^5+2 z a^5+2 z^{10} a^4+7 z^8 a^4-39 z^6 a^4+44 z^4 a^4-14 z^2 a^4+5 z^9 a^3-15 z^7 a^3+11 z^5 a^3-z^3 a^3+2 z a^3+4 z^8 a^2-15 z^6 a^2+17 z^4 a^2-4 z^2 a^2-2 a^2+z^7 a-3 z^5 a+2 z^3 a
The A2 invariant q^{30}-q^{28}+2 q^{24}-3 q^{22}+3 q^{20}-2 q^{18}-q^{16}+q^{14}-5 q^{12}+4 q^{10}-2 q^8+3 q^6+3 q^4-q^2+2- q^{-2}
The G2 invariant Data:K11a261/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 -64 0 0 \frac{352}{3} -\frac{128}{3} 16 0 128 0 0 160 \frac{320}{3} 192 -32 32

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=-4 is the signature of K11a261. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3           1-1
1          3 3
-1         41 -3
-3        93  6
-5       95   -4
-7      118    3
-9     109     -1
-11    811      -3
-13   610       4
-15  38        -5
-17 16         5
-19 3          -3
-211           1
Integral Khovanov Homology

(db, data source)

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