From Knot Atlas
Jump to: navigation, search






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

Visit K11a121 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,5,15,6 X18,7,19,8 X12,10,13,9 X2,11,3,12 X22,13,1,14 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, -5, 7, -3, 8, -11, 9, -4, 10, -9, 11, -7
Dowker-Thistlethwaite code 4 10 14 18 12 2 22 8 20 6 16
A Braid Representative
A Morse Link Presentation K11a121 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:K11a121/ThurstonBennequinNumber
Hyperbolic Volume 14.7555
A-Polynomial See Data:K11a121/A-polynomial

[edit Notes for K11a121'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 K11a121's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-9 t^2+29 t-41+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 { 119, -2 }
Jones polynomial q^3-4 q^2+8 q-12+17 q^{-1} -19 q^{-2} +19 q^{-3} -16 q^{-4} +12 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^8+3 z^2 a^6+2 a^6-3 z^4 a^4-4 z^2 a^4-2 a^4+z^6 a^2+2 z^4 a^2+4 z^2 a^2+2 a^2-2 z^4-2 z^2+z^2 a^{-2}
Kauffman polynomial (db, data sources) a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+7 a^3 z^9+4 a z^9+5 a^6 z^8+10 a^4 z^8+11 a^2 z^8+6 z^8+5 a^7 z^7+7 a^5 z^7-3 a^3 z^7-a z^7+4 z^7 a^{-1} +3 a^8 z^6-2 a^6 z^6-15 a^4 z^6-25 a^2 z^6+z^6 a^{-2} -14 z^6+a^9 z^5-7 a^7 z^5-19 a^5 z^5-13 a^3 z^5-12 a z^5-10 z^5 a^{-1} -5 a^8 z^4-6 a^6 z^4+11 a^2 z^4-2 z^4 a^{-2} +8 z^4-2 a^9 z^3+4 a^7 z^3+14 a^5 z^3+9 a^3 z^3+7 a z^3+6 z^3 a^{-1} +3 a^8 z^2+7 a^6 z^2+7 a^4 z^2+2 a^2 z^2+z^2 a^{-2} +a^9 z-a^7 z-3 a^5 z-a^3 z-a^8-2 a^6-2 a^4-2 a^2
The A2 invariant Data:K11a121/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a121/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a66,}

Vassiliev invariants

V2 and V3: (2, -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
8 -32 32 \frac{460}{3} \frac{116}{3} -256 -\frac{2144}{3} -\frac{224}{3} -224 \frac{256}{3} 512 \frac{3680}{3} \frac{928}{3} \frac{49111}{15} -\frac{11884}{15} \frac{105724}{45} \frac{1145}{9} \frac{6151}{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=-2 is the signature of K11a121. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          3 -3
3         51 4
1        73  -4
-1       105   5
-3      108    -2
-5     99     0
-7    710      3
-9   59       -4
-11  27        5
-13 15         -4
-15 2          2
-171           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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.