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

Visit K11a282 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X16,5,17,6 X14,8,15,7 X4,10,5,9 X18,11,19,12 X20,13,21,14 X22,16,1,15 X2,17,3,18 X12,19,13,20 X8,21,9,22
Gauss code 1, -9, 2, -5, 3, -1, 4, -11, 5, -2, 6, -10, 7, -4, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 6 10 16 14 4 18 20 22 2 12 8
A Braid Representative
A Morse Link Presentation K11a282 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a282/ThurstonBennequinNumber
Hyperbolic Volume 16.6494
A-Polynomial See Data:K11a282/A-polynomial

[edit Notes for K11a282's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a282's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-16 t^2+26 t-29+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 { 127, -2 }
Jones polynomial -q^4+4 q^3-8 q^2+13 q-17+20 q^{-1} -20 q^{-2} +18 q^{-3} -13 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+a^4 z^6-5 a^2 z^6+2 z^6+3 a^4 z^4-9 a^2 z^4-z^4 a^{-2} +7 z^4+2 a^4 z^2-6 a^2 z^2-2 z^2 a^{-2} +6 z^2+1
Kauffman polynomial (db, data sources) 3 a^2 z^{10}+3 z^{10}+9 a^3 z^9+15 a z^9+6 z^9 a^{-1} +12 a^4 z^8+10 a^2 z^8+4 z^8 a^{-2} +2 z^8+11 a^5 z^7-14 a^3 z^7-46 a z^7-20 z^7 a^{-1} +z^7 a^{-3} +8 a^6 z^6-20 a^4 z^6-47 a^2 z^6-14 z^6 a^{-2} -33 z^6+4 a^7 z^5-13 a^5 z^5+a^3 z^5+39 a z^5+18 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-7 a^6 z^4+10 a^4 z^4+45 a^2 z^4+14 z^4 a^{-2} +41 z^4-2 a^7 z^3+3 a^5 z^3+3 a^3 z^3-9 a z^3-5 z^3 a^{-1} +2 z^3 a^{-3} +a^6 z^2-2 a^4 z^2-12 a^2 z^2-5 z^2 a^{-2} -14 z^2+1
The A2 invariant q^{20}-2 q^{18}+2 q^{16}-2 q^{14}-q^{12}+3 q^{10}-3 q^8+5 q^6-2 q^4+q^2+1-3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12}
The G2 invariant Data:K11a282/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a81,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a81,}

Vassiliev invariants

V2 and V3: (0, 0)
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 0 0 0 0 0 0 -64 64 0 0 0 0 0 160 -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=-2 is the signature of K11a282. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9           1-1
7          3 3
5         51 -4
3        83  5
1       95   -4
-1      118    3
-3     1010     0
-5    810      -2
-7   510       5
-9  38        -5
-11 15         4
-13 3          -3
-151           1
Integral Khovanov Homology

(db, data source)

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