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

Visit K11a257 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X8493 X14,5,15,6 X2837 X16,10,17,9 X18,11,19,12 X20,13,21,14 X4,15,5,16 X22,18,1,17 X12,19,13,20 X10,21,11,22
Gauss code 1, -4, 2, -8, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code 6 8 14 2 16 18 20 4 22 12 10
A Braid Representative
A Morse Link Presentation K11a257 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a257/ThurstonBennequinNumber
Hyperbolic Volume 13.7301
A-Polynomial See Data:K11a257/A-polynomial

[edit Notes for K11a257's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a257's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+12 t^2-19 t+23-19 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 97, 0 }
Jones polynomial -q^5+3 q^4-6 q^3+10 q^2-13 q+16-15 q^{-1} +13 q^{-2} -10 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) z^8-2 a^2 z^6-z^6 a^{-2} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +14 z^4+3 a^4 z^2-13 a^2 z^2-5 z^2 a^{-2} +15 z^2+2 a^4-6 a^2-2 a^{-2} +7
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+3 a^3 z^9+7 a z^9+4 z^9 a^{-1} +4 a^4 z^8+7 a^2 z^8+6 z^8 a^{-2} +9 z^8+3 a^5 z^7-3 a^3 z^7-17 a z^7-6 z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-10 a^4 z^6-28 a^2 z^6-16 z^6 a^{-2} +3 z^6 a^{-4} -36 z^6-9 a^5 z^5-7 a^3 z^5+14 a z^5-11 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+6 a^4 z^4+38 a^2 z^4+20 z^4 a^{-2} -6 z^4 a^{-4} +55 z^4+7 a^5 z^3+8 a^3 z^3+2 a z^3+10 z^3 a^{-1} +7 z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-4 a^4 z^2-25 a^2 z^2-11 z^2 a^{-2} +z^2 a^{-4} -31 z^2-2 a^5 z-4 a^3 z-4 a z-4 z a^{-1} -2 z a^{-3} +2 a^4+6 a^2+2 a^{-2} +7
The A2 invariant q^{18}+q^{12}-3 q^{10}+q^8-2 q^6-q^4+3 q^2-1+5 q^{-2} - q^{-4} + q^{-6} + q^{-8} -2 q^{-10} + q^{-12} - q^{-14}
The G2 invariant Data:K11a257/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_118,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a110,}

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 -32 -16 0 -\frac{32}{3} -\frac{224}{3} 48 0 128 0 0 272 -80 \frac{880}{3} -\frac{176}{3} 48

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=0 is the signature of K11a257. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           1-1
9          2 2
7         41 -3
5        62  4
3       74   -3
1      96    3
-1     78     1
-3    68      -2
-5   47       3
-7  26        -4
-9 14         3
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

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