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

Visit K11a306 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X12,3,13,4 X14,5,15,6 X16,8,17,7 X18,9,19,10 X20,11,21,12 X4,13,5,14 X2,15,3,16 X22,18,1,17 X10,19,11,20 X8,21,9,22
Gauss code 1, -8, 2, -7, 3, -1, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code 6 12 14 16 18 20 4 2 22 10 8
A Braid Representative
A Morse Link Presentation K11a306 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a306/ThurstonBennequinNumber
Hyperbolic Volume 13.5238
A-Polynomial See Data:K11a306/A-polynomial

[edit Notes for K11a306'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 K11a306's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+13 t^2-21 t+25-21 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+3 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 105, -4 }
Jones polynomial -q+3-5 q^{-1} +10 q^{-2} -13 q^{-3} +16 q^{-4} -17 q^{-5} +15 q^{-6} -12 q^{-7} +8 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^6-8 z^4 a^6-10 z^2 a^6-4 a^6+z^8 a^4+6 z^6 a^4+14 z^4 a^4+14 z^2 a^4+4 a^4-z^6 a^2-4 z^4 a^2-4 z^2 a^2
Kauffman polynomial (db, data sources) z^4 a^{12}+4 z^5 a^{11}-2 z^3 a^{11}+8 z^6 a^{10}-8 z^4 a^{10}+2 z^2 a^{10}+10 z^7 a^9-13 z^5 a^9+6 z^3 a^9-z a^9+8 z^8 a^8-7 z^6 a^8-3 z^4 a^8+z^2 a^8+a^8+4 z^9 a^7+5 z^7 a^7-23 z^5 a^7+15 z^3 a^7-3 z a^7+z^{10} a^6+11 z^8 a^6-32 z^6 a^6+27 z^4 a^6-14 z^2 a^6+4 a^6+7 z^9 a^5-15 z^7 a^5+4 z^5 a^5+3 z^3 a^5-z a^5+z^{10} a^4+6 z^8 a^4-30 z^6 a^4+39 z^4 a^4-21 z^2 a^4+4 a^4+3 z^9 a^3-9 z^7 a^3+6 z^5 a^3+z a^3+3 z^8 a^2-13 z^6 a^2+18 z^4 a^2-8 z^2 a^2+z^7 a-4 z^5 a+4 z^3 a
The A2 invariant q^{30}-q^{28}+q^{24}-3 q^{22}+2 q^{20}-2 q^{18}+q^{14}-3 q^{12}+4 q^{10}-q^8+3 q^6+2 q^4-q^2+1- q^{-2}
The G2 invariant Data:K11a306/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a175,}

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

Vassiliev invariants

V2 and V3: (2, -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
8 -16 32 \frac{76}{3} -\frac{28}{3} -128 -\frac{64}{3} -\frac{160}{3} 112 \frac{256}{3} 128 \frac{608}{3} -\frac{224}{3} -\frac{1289}{15} \frac{10276}{15} -\frac{37556}{45} -\frac{1159}{9} -\frac{2009}{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=-4 is the signature of K11a306. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3           1-1
1          2 2
-1         31 -2
-3        72  5
-5       74   -3
-7      96    3
-9     87     -1
-11    79      -2
-13   58       3
-15  37        -4
-17 15         4
-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}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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