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

Visit K11a262 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X8394 X14,5,15,6 X16,8,17,7 X4,9,5,10 X20,11,21,12 X18,13,19,14 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, -7, 10, -6, 11, -9
Dowker-Thistlethwaite code 6 8 14 16 4 20 18 2 22 12 10
A Braid Representative
A Morse Link Presentation K11a262 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a262/ThurstonBennequinNumber
Hyperbolic Volume 14.8139
A-Polynomial See Data:K11a262/A-polynomial

[edit Notes for K11a262'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 K11a262's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-11 t^2+25 t-31+25 t^{-1} -11 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 107, -2 }
Jones polynomial -q^2+4 q-7+12 q^{-1} -15 q^{-2} +17 q^{-3} -17 q^{-4} +14 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+2 z^2 a^4+a^4+z^6 a^2+2 z^4 a^2+z^2 a^2-z^4-z^2+1
Kauffman polynomial (db, data sources) z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+3 z^7 a^9-9 z^5 a^9+5 z^3 a^9-z a^9+5 z^8 a^8-16 z^6 a^8+15 z^4 a^8-7 z^2 a^8+a^8+5 z^9 a^7-15 z^7 a^7+15 z^5 a^7-6 z^3 a^7+z a^7+2 z^{10} a^6+3 z^8 a^6-25 z^6 a^6+39 z^4 a^6-18 z^2 a^6+2 a^6+10 z^9 a^5-32 z^7 a^5+43 z^5 a^5-21 z^3 a^5+4 z a^5+2 z^{10} a^4+4 z^8 a^4-19 z^6 a^4+27 z^4 a^4-12 z^2 a^4+a^4+5 z^9 a^3-8 z^7 a^3+9 z^5 a^3-8 z^3 a^3+2 z a^3+6 z^8 a^2-7 z^6 a^2-z^4 a^2+6 z^7 a-9 z^5 a+z^3 a+4 z^6-7 z^4+2 z^2+1+z^5 a^{-1} -z^3 a^{-1}
The A2 invariant Data:K11a262/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a262/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a10,}

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

Vassiliev invariants

V2 and V3: (-1, 3)
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
-4 24 8 -\frac{206}{3} -\frac{10}{3} -96 16 -64 24 -\frac{32}{3} 288 \frac{824}{3} \frac{40}{3} \frac{23009}{30} \frac{2342}{15} \frac{14938}{45} -\frac{737}{18} \frac{1409}{30}

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 K11a262. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5           1-1
3          3 3
1         41 -3
-1        83  5
-3       85   -3
-5      97    2
-7     88     0
-9    69      -3
-11   48       4
-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}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
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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