From Knot Atlas
Jump to: navigation, search






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

Visit K11a155 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X18,5,19,6 X12,7,13,8 X2,10,3,9 X16,11,17,12 X20,13,21,14 X8,15,9,16 X22,17,1,18 X14,19,15,20 X6,21,7,22
Gauss code 1, -5, 2, -1, 3, -11, 4, -8, 5, -2, 6, -4, 7, -10, 8, -6, 9, -3, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 10 18 12 2 16 20 8 22 14 6
A Braid Representative
A Morse Link Presentation K11a155 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a155/ThurstonBennequinNumber
Hyperbolic Volume 18.5384
A-Polynomial See Data:K11a155/A-polynomial

[edit Notes for K11a155'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 K11a155's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-16 t^2+40 t-53+40 t^{-1} -16 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+2 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 171, -2 }
Jones polynomial -q^2+5 q-12+19 q^{-1} -24 q^{-2} +29 q^{-3} -27 q^{-4} +23 q^{-5} -17 q^{-6} +9 q^{-7} -4 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-3 z^4 a^6-6 z^2 a^6-5 a^6+2 z^6 a^4+6 z^4 a^4+10 z^2 a^4+6 a^4+z^6 a^2-2 z^2 a^2-a^2-z^4
Kauffman polynomial (db, data sources) z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-9 z^5 a^9+7 z^3 a^9-2 z a^9+7 z^8 a^8-13 z^6 a^8+7 z^4 a^8-2 z^2 a^8+a^8+7 z^9 a^7-5 z^7 a^7-14 z^5 a^7+20 z^3 a^7-9 z a^7+3 z^{10} a^6+15 z^8 a^6-45 z^6 a^6+38 z^4 a^6-15 z^2 a^6+5 a^6+18 z^9 a^5-25 z^7 a^5-7 z^5 a^5+22 z^3 a^5-10 z a^5+3 z^{10} a^4+24 z^8 a^4-61 z^6 a^4+45 z^4 a^4-16 z^2 a^4+6 a^4+11 z^9 a^3-4 z^7 a^3-18 z^5 a^3+13 z^3 a^3-4 z a^3+16 z^8 a^2-25 z^6 a^2+13 z^4 a^2-4 z^2 a^2+a^2+12 z^7 a-15 z^5 a+4 z^3 a-z a+5 z^6-3 z^4+z^5 a^{-1}
The A2 invariant Data:K11a155/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a155/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (3, -4)
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
12 -32 72 126 18 -384 -\frac{1664}{3} -\frac{320}{3} -64 288 512 1512 216 \frac{24751}{10} \frac{738}{5} \frac{14062}{15} -\frac{335}{6} \frac{1231}{10}

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 K11a155. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5           1-1
3          4 4
1         81 -7
-1        114  7
-3       149   -5
-5      1510    5
-7     1214     2
-9    1115      -4
-11   612       6
-13  311        -8
-15 16         5
-17 3          -3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-3 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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).

Back to the top.