From Knot Atlas
Jump to: navigation, search






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

Visit K11a52 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X14,5,15,6 X12,8,13,7 X2,9,3,10 X18,12,19,11 X22,13,1,14 X20,16,21,15 X10,18,11,17 X16,20,17,19 X6,21,7,22
Gauss code 1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -4, 7, -3, 8, -10, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 8 14 12 2 18 22 20 10 16 6
A Braid Representative
A Morse Link Presentation K11a52 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a52/ThurstonBennequinNumber
Hyperbolic Volume 16.8243
A-Polynomial See Data:K11a52/A-polynomial

[edit Notes for K11a52's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a52's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+13 t^2-32 t+43-32 t^{-1} +13 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6+z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 137, 0 }
Jones polynomial q^6-5 q^5+10 q^4-15 q^3+20 q^2-22 q+22-18 q^{-1} +13 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6+2 a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -z^4-a^4 z^2+3 a^2 z^2+z^2 a^{-2} -z^2-a^4+2 a^2+2 a^{-2} - a^{-4} -1
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10}+5 a z^9+12 z^9 a^{-1} +7 z^9 a^{-3} +6 a^2 z^8+14 z^8 a^{-2} +9 z^8 a^{-4} +11 z^8+5 a^3 z^7-18 z^7 a^{-1} -8 z^7 a^{-3} +5 z^7 a^{-5} +3 a^4 z^6-5 a^2 z^6-41 z^6 a^{-2} -21 z^6 a^{-4} +z^6 a^{-6} -27 z^6+a^5 z^5-6 a^3 z^5-9 a z^5-z^5 a^{-1} -9 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4-a^2 z^4+27 z^4 a^{-2} +11 z^4 a^{-4} -z^4 a^{-6} +19 z^4-2 a^5 z^3+2 a^3 z^3+10 a z^3+10 z^3 a^{-1} +7 z^3 a^{-3} +3 z^3 a^{-5} +3 a^4 z^2+4 a^2 z^2-2 z^2 a^{-2} -z^2+a^5 z-3 a z-3 z a^{-1} +z a^{-5} -a^4-2 a^2-2 a^{-2} - a^{-4} -1
The A2 invariant Data:K11a52/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a52/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: (2, 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
8 0 32 \frac{172}{3} \frac{20}{3} 0 0 32 -32 \frac{256}{3} 0 \frac{1376}{3} \frac{160}{3} \frac{8551}{15} \frac{1156}{15} \frac{5404}{45} \frac{377}{9} -\frac{89}{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=0 is the signature of K11a52. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13           11
11          4 -4
9         61 5
7        94  -5
5       116   5
3      119    -2
1     1111     0
-1    812      4
-3   510       -5
-5  28        6
-7 15         -4
-9 2          2
-111           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\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.