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

Visit K11a253 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X8493 X12,5,13,6 X2837 X18,9,19,10 X20,12,21,11 X4,13,5,14 X10,15,11,16 X22,17,1,18 X14,20,15,19 X16,21,17,22
Gauss code 1, -4, 2, -7, 3, -1, 4, -2, 5, -8, 6, -3, 7, -10, 8, -11, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code 6 8 12 2 18 20 4 10 22 14 16
A Braid Representative
A Morse Link Presentation K11a253 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a253/ThurstonBennequinNumber
Hyperbolic Volume 16.9235
A-Polynomial See Data:K11a253/A-polynomial

[edit Notes for K11a253'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 K11a253's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+16 t^2-27 t+33-27 t^{-1} +16 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 133, 0 }
Jones polynomial -q^5+4 q^4-9 q^3+15 q^2-19 q+22-21 q^{-1} +18 q^{-2} -13 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-8 a^2 z^4-3 z^4 a^{-2} +10 z^4+3 a^4 z^2-11 a^2 z^2-3 z^2 a^{-2} +10 z^2+2 a^4-5 a^2- a^{-2} +5
Kauffman polynomial (db, data sources) 2 a^2 z^{10}+2 z^{10}+5 a^3 z^9+12 a z^9+7 z^9 a^{-1} +5 a^4 z^8+9 a^2 z^8+10 z^8 a^{-2} +14 z^8+3 a^5 z^7-8 a^3 z^7-24 a z^7-5 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-12 a^4 z^6-34 a^2 z^6-17 z^6 a^{-2} +4 z^6 a^{-4} -42 z^6-8 a^5 z^5+a^3 z^5+16 a z^5-6 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+10 a^4 z^4+42 a^2 z^4+12 z^4 a^{-2} -5 z^4 a^{-4} +46 z^4+6 a^5 z^3+4 a^3 z^3-a z^3+7 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +2 a^6 z^2-6 a^4 z^2-24 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} -23 z^2-2 a^5 z-3 a^3 z-2 a z-2 z a^{-1} -z a^{-3} +2 a^4+5 a^2+ a^{-2} +5
The A2 invariant q^{18}+2 q^{12}-4 q^{10}+2 q^8-2 q^6-2 q^4+4 q^2-3+6 q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -3 q^{-10} +2 q^{-12} - q^{-14}
The G2 invariant Data:K11a253/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a251,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a228, K11a251,}

Vassiliev invariants

V2 and V3: (-1, 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
-4 16 8 -\frac{14}{3} \frac{14}{3} -64 -\frac{416}{3} -\frac{224}{3} -16 -\frac{32}{3} 128 \frac{56}{3} -\frac{56}{3} \frac{11009}{30} -\frac{1378}{15} \frac{13378}{45} -\frac{65}{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=0 is the signature of K11a253. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           1-1
9          3 3
7         61 -5
5        93  6
3       106   -4
1      129    3
-1     1011     1
-3    811      -3
-5   510       5
-7  28        -6
-9 15         4
-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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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