From Knot Atlas
Jump to: navigation, search






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

Visit K11a96 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X18,7,19,8 X14,9,15,10 X2,11,3,12 X22,14,1,13 X8,15,9,16 X20,18,21,17 X6,19,7,20 X16,22,17,21
Gauss code 1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -5, 8, -11, 9, -4, 10, -9, 11, -7
Dowker-Thistlethwaite code 4 10 12 18 14 2 22 8 20 6 16
A Braid Representative
A Morse Link Presentation K11a96 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a96/ThurstonBennequinNumber
Hyperbolic Volume 14.8744
A-Polynomial See Data:K11a96/A-polynomial

[edit Notes for K11a96's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a96's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+9 t^2-29 t+43-29 t^{-1} +9 t^{-2} - t^{-3}
Conway polynomial -z^6+3 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 121, 0 }
Jones polynomial -q^5+4 q^4-8 q^3+13 q^2-17 q+20-19 q^{-1} +16 q^{-2} -12 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-3 z^2 a^4-2 a^4+3 z^4 a^2+4 z^2 a^2+2 a^2-z^6-2 z^4-4 z^2-1+2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} -z^2 a^{-4}
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+9 a^2 z^8+7 z^8 a^{-2} +12 z^8+3 a^5 z^7+2 a^3 z^7-3 a z^7+5 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-6 a^4 z^6-17 a^2 z^6-7 z^6 a^{-2} +4 z^6 a^{-4} -21 z^6-8 a^5 z^5-15 a^3 z^5-11 a z^5-16 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4-3 a^4 z^4+2 a^2 z^4-z^4 a^{-2} -6 z^4 a^{-4} +7 z^4+7 a^5 z^3+11 a^3 z^3+6 a z^3+8 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +3 a^6 z^2+7 a^4 z^2+6 a^2 z^2+3 z^2 a^{-2} +2 z^2 a^{-4} +3 z^2-2 a^5 z-2 a^3 z-a^6-2 a^4-2 a^2- a^{-2} -1
The A2 invariant Data:K11a96/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a96/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a28, K11a87,}

Vassiliev invariants

V2 and V3: (-2, 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
-8 16 32 -\frac{28}{3} -\frac{20}{3} -128 -\frac{320}{3} -\frac{224}{3} 48 -\frac{256}{3} 128 \frac{224}{3} \frac{160}{3} \frac{5729}{15} \frac{524}{15} \frac{9716}{45} -\frac{449}{9} \frac{449}{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 K11a96. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           1-1
9          3 3
7         51 -4
5        83  5
3       95   -4
1      118    3
-1     910     1
-3    710      -3
-5   59       4
-7  27        -5
-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}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
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).

Back to the top.