From Knot Atlas
Jump to: navigation, search






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

Visit K11a307 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X12,3,13,4 X14,5,15,6 X16,8,17,7 X20,9,21,10 X18,11,19,12 X4,13,5,14 X2,15,3,16 X22,18,1,17 X10,19,11,20 X8,21,9,22
Gauss code 1, -8, 2, -7, 3, -1, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -6, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 12 14 16 20 18 4 2 22 10 8
A Braid Representative
A Morse Link Presentation K11a307 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:K11a307/ThurstonBennequinNumber
Hyperbolic Volume 11.9838
A-Polynomial See Data:K11a307/A-polynomial

[edit Notes for K11a307'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 K11a307's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-9 t^2+19 t-23+19 t^{-1} -9 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 83, -2 }
Jones polynomial -q^2+3 q-5+9 q^{-1} -11 q^{-2} +13 q^{-3} -13 q^{-4} +11 q^{-5} -8 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-2 z^4 a^6-5 z^2 a^6-3 a^6+z^6 a^4+3 z^4 a^4+4 z^2 a^4+2 a^4+z^6 a^2+3 z^4 a^2+3 z^2 a^2+a^2-z^4-2 z^2
Kauffman polynomial (db, data sources) z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+7 z^3 a^9-z a^9+4 z^8 a^8-13 z^6 a^8+11 z^4 a^8-4 z^2 a^8+a^8+3 z^9 a^7-7 z^7 a^7+2 z^5 a^7+2 z^3 a^7-z a^7+z^{10} a^6+4 z^8 a^6-22 z^6 a^6+33 z^4 a^6-18 z^2 a^6+3 a^6+6 z^9 a^5-20 z^7 a^5+29 z^5 a^5-15 z^3 a^5+2 z a^5+z^{10} a^4+4 z^8 a^4-19 z^6 a^4+32 z^4 a^4-17 z^2 a^4+2 a^4+3 z^9 a^3-6 z^7 a^3+8 z^5 a^3-5 z^3 a^3+2 z a^3+4 z^8 a^2-8 z^6 a^2+6 z^4 a^2-z^2 a^2-a^2+4 z^7 a-8 z^5 a+3 z^3 a+3 z^6-7 z^4+3 z^2+z^5 a^{-1} -2 z^3 a^{-1}
The A2 invariant Data:K11a307/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a307/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_83, K11a323,}

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

Vassiliev invariants

V2 and V3: (1, 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
4 0 8 -\frac{130}{3} -\frac{62}{3} 0 224 -32 160 \frac{32}{3} 0 -\frac{520}{3} -\frac{248}{3} -\frac{23249}{30} \frac{10178}{15} -\frac{45538}{45} -\frac{2863}{18} -\frac{5009}{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=-2 is the signature of K11a307. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5           1-1
3          2 2
1         31 -2
-1        62  4
-3       64   -2
-5      75    2
-7     66     0
-9    57      -2
-11   36       3
-13  25        -3
-15 13         2
-17 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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.