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

Visit K11a225 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X18,5,19,6 X20,7,21,8 X16,10,17,9 X14,12,15,11 X2,13,3,14 X10,16,11,15 X22,17,1,18 X6,19,7,20 X8,21,9,22
Gauss code 1, -7, 2, -1, 3, -10, 4, -11, 5, -8, 6, -2, 7, -6, 8, -5, 9, -3, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 12 18 20 16 14 2 10 22 6 8
A Braid Representative
A Morse Link Presentation K11a225 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:K11a225/ThurstonBennequinNumber
Hyperbolic Volume 9.36224
A-Polynomial See Data:K11a225/A-polynomial

[edit Notes for K11a225'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 K11a225's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+8 t^2-11 t+11-11 t^{-1} +8 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-4 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 53, -4 }
Jones polynomial q^2-2 q+3-5 q^{-1} +7 q^{-2} -7 q^{-3} +8 q^{-4} -7 q^{-5} +6 q^{-6} -4 q^{-7} +2 q^{-8} - q^{-9}
HOMFLY-PT polynomial (db, data sources) -z^2 a^8-2 a^8+2 z^4 a^6+6 z^2 a^6+3 a^6-z^6 a^4-3 z^4 a^4-z^2 a^4-z^6 a^2-4 z^4 a^2-4 z^2 a^2-a^2+z^4+3 z^2+1
Kauffman polynomial (db, data sources) z^3 a^{11}-z a^{11}+2 z^4 a^{10}-z^2 a^{10}+3 z^5 a^9-2 z^3 a^9+z a^9+4 z^6 a^8-6 z^4 a^8+5 z^2 a^8-2 a^8+4 z^7 a^7-7 z^5 a^7+2 z^3 a^7+z a^7+3 z^8 a^6-4 z^6 a^6-7 z^4 a^6+9 z^2 a^6-3 a^6+2 z^9 a^5-3 z^7 a^5-5 z^5 a^5+4 z^3 a^5+z^{10} a^4-z^8 a^4-5 z^6 a^4+3 z^4 a^4+z^2 a^4+4 z^9 a^3-19 z^7 a^3+28 z^5 a^3-16 z^3 a^3+3 z a^3+z^{10} a^2-3 z^8 a^2-3 z^6 a^2+13 z^4 a^2-8 z^2 a^2+a^2+2 z^9 a-12 z^7 a+23 z^5 a-15 z^3 a+2 z a+z^8-6 z^6+11 z^4-6 z^2+1
The A2 invariant Data:K11a225/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a225/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: (3, -8)
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
12 -64 72 318 66 -768 -\frac{5248}{3} -\frac{736}{3} -384 288 2048 3816 792 \frac{98991}{10} -\frac{3822}{5} \frac{77182}{15} \frac{1201}{6} \frac{7631}{10}

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 K11a225. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5           11
3          1 -1
1         21 1
-1        31  -2
-3       42   2
-5      44    0
-7     43     1
-9    34      1
-11   34       -1
-13  13        2
-15 13         -2
-17 1          1
-191           -1
Integral Khovanov Homology

(db, data source)

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

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