From Knot Atlas
Jump to: navigation, search






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

Visit K11a207 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X16,5,17,6 X18,7,19,8 X20,9,21,10 X14,12,15,11 X2,13,3,14 X22,15,1,16 X6,17,7,18 X10,19,11,20 X8,21,9,22
Gauss code 1, -7, 2, -1, 3, -9, 4, -11, 5, -10, 6, -2, 7, -6, 8, -3, 9, -4, 10, -5, 11, -8
Dowker-Thistlethwaite code 4 12 16 18 20 14 2 22 6 10 8
A Braid Representative
A Morse Link Presentation K11a207 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a207/ThurstonBennequinNumber
Hyperbolic Volume 12.7356
A-Polynomial See Data:K11a207/A-polynomial

[edit Notes for K11a207's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a207's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-19 t+21-19 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4+7 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 85, -4 }
Jones polynomial 1-3 q^{-1} +6 q^{-2} -9 q^{-3} +12 q^{-4} -13 q^{-5} +14 q^{-6} -11 q^{-7} +8 q^{-8} -5 q^{-9} +2 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-2 a^{10}+2 z^4 a^8+5 z^2 a^8+2 a^8-z^6 a^6-2 z^4 a^6-z^6 a^4-2 z^4 a^4+z^2 a^4+a^4+z^4 a^2+2 z^2 a^2
Kauffman polynomial (db, data sources) z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+3 z^7 a^{11}-6 z^5 a^{11}+5 z^3 a^{11}-3 z a^{11}+3 z^8 a^{10}-5 z^6 a^{10}+6 z^4 a^{10}-5 z^2 a^{10}+2 a^{10}+2 z^9 a^9-5 z^5 a^9+10 z^3 a^9-4 z a^9+z^{10} a^8+2 z^8 a^8-5 z^6 a^8+6 z^4 a^8-2 z^2 a^8+2 a^8+5 z^9 a^7-12 z^7 a^7+15 z^5 a^7-9 z^3 a^7+2 z a^7+z^{10} a^6+3 z^8 a^6-10 z^6 a^6+7 z^4 a^6-2 z^2 a^6+3 z^9 a^5-6 z^7 a^5+4 z^5 a^5-5 z^3 a^5+z a^5+4 z^8 a^4-11 z^6 a^4+8 z^4 a^4-4 z^2 a^4+a^4+3 z^7 a^3-9 z^5 a^3+6 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^2
The A2 invariant Data:K11a207/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a207/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: (7, -19)
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
28 -152 392 \frac{3218}{3} \frac{598}{3} -4256 -\frac{24560}{3} -\frac{4256}{3} -1400 \frac{10976}{3} 11552 \frac{90104}{3} \frac{16744}{3} \frac{1886857}{30} -\frac{26474}{15} \frac{1334954}{45} \frac{10871}{18} \frac{128617}{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=-4 is the signature of K11a207. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
1           11
-1          2 -2
-3         41 3
-5        63  -3
-7       63   3
-9      76    -1
-11     76     1
-13    47      3
-15   47       -3
-17  14        3
-19 14         -3
-21 1          1
-231           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\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.