K11a35

K11a34

K11a36

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a35 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X16,9,17,10 X18,12,19,11 X22,13,1,14 X6,15,7,16 X20,18,21,17 X12,20,13,19 X10,21,11,22
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, -9, 11, -7
Dowker-Thistlethwaite code	4 8 14 2 16 18 22 6 20 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a35/ThurstonBennequinNumber Hyperbolic Volume 15.3717 A-Polynomial See Data:K11a35/A-polynomial

[edit Notes for K11a35's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ [0,4] }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a35's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+14 t^2-25 t+31-25 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+4 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 121, 0 }
Jones polynomial	[math]\displaystyle{ q^6-4 q^5+8 q^4-13 q^3+17 q^2-19 q+20-16 q^{-1} +12 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +15 z^4-6 a^2 z^2-11 z^2 a^{-2} +2 z^2 a^{-4} +17 z^2-3 a^2-5 a^{-2} + a^{-4} +8 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-2} +z^{10}+4 a z^9+8 z^9 a^{-1} +4 z^9 a^{-3} +6 a^2 z^8+14 z^8 a^{-2} +6 z^8 a^{-4} +14 z^8+5 a^3 z^7+2 a z^7-5 z^7 a^{-1} +2 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-9 a^2 z^6-41 z^6 a^{-2} -13 z^6 a^{-4} +z^6 a^{-6} -39 z^6+a^5 z^5-7 a^3 z^5-14 a z^5-22 z^5 a^{-1} -26 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+10 a^2 z^4+39 z^4 a^{-2} +6 z^4 a^{-4} -2 z^4 a^{-6} +46 z^4-2 a^5 z^3+3 a^3 z^3+17 a z^3+31 z^3 a^{-1} +26 z^3 a^{-3} +7 z^3 a^{-5} +2 a^4 z^2-9 a^2 z^2-19 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -27 z^2+a^5 z-a^3 z-7 a z-11 z a^{-1} -8 z a^{-3} -2 z a^{-5} +3 a^2+5 a^{-2} + a^{-4} +8 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+q^{12}-3 q^{10}+q^8+q^6-2 q^4+6 q^2-1+4 q^{-2} -2 q^{-6} +2 q^{-8} -4 q^{-10} + q^{-12} - q^{-16} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+9 q^{72}-8 q^{70}+q^{68}+13 q^{66}-29 q^{64}+47 q^{62}-59 q^{60}+52 q^{58}-29 q^{56}-18 q^{54}+82 q^{52}-145 q^{50}+189 q^{48}-190 q^{46}+123 q^{44}-q^{42}-162 q^{40}+313 q^{38}-397 q^{36}+369 q^{34}-220 q^{32}-26 q^{30}+284 q^{28}-458 q^{26}+482 q^{24}-334 q^{22}+71 q^{20}+198 q^{18}-370 q^{16}+362 q^{14}-175 q^{12}-93 q^{10}+339 q^8-419 q^6+293 q^4+9 q^2-356+618 q^{-2} -666 q^{-4} +474 q^{-6} -91 q^{-8} -335 q^{-10} +668 q^{-12} -772 q^{-14} +624 q^{-16} -278 q^{-18} -136 q^{-20} +452 q^{-22} -573 q^{-24} +464 q^{-26} -186 q^{-28} -133 q^{-30} +350 q^{-32} -382 q^{-34} +211 q^{-36} +72 q^{-38} -347 q^{-40} +480 q^{-42} -419 q^{-44} +178 q^{-46} +137 q^{-48} -410 q^{-50} +541 q^{-52} -482 q^{-54} +277 q^{-56} -11 q^{-58} -226 q^{-60} +352 q^{-62} -351 q^{-64} +255 q^{-66} -106 q^{-68} -28 q^{-70} +113 q^{-72} -139 q^{-74} +116 q^{-76} -69 q^{-78} +26 q^{-80} +7 q^{-82} -22 q^{-84} +21 q^{-86} -16 q^{-88} +8 q^{-90} -3 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

K11a35 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a35"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ t^4-5 t^3+14 t^2-25 t+31-25 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+3 z^6+4 z^4+2 z^2+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 121, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ q^6-4 q^5+8 q^4-13 q^3+17 q^2-19 q+20-16 q^{-1} +12 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +15 z^4-6 a^2 z^2-11 z^2 a^{-2} +2 z^2 a^{-4} +17 z^2-3 a^2-5 a^{-2} + a^{-4} +8 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^{10} a^{-2} +z^{10}+4 a z^9+8 z^9 a^{-1} +4 z^9 a^{-3} +6 a^2 z^8+14 z^8 a^{-2} +6 z^8 a^{-4} +14 z^8+5 a^3 z^7+2 a z^7-5 z^7 a^{-1} +2 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-9 a^2 z^6-41 z^6 a^{-2} -13 z^6 a^{-4} +z^6 a^{-6} -39 z^6+a^5 z^5-7 a^3 z^5-14 a z^5-22 z^5 a^{-1} -26 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+10 a^2 z^4+39 z^4 a^{-2} +6 z^4 a^{-4} -2 z^4 a^{-6} +46 z^4-2 a^5 z^3+3 a^3 z^3+17 a z^3+31 z^3 a^{-1} +26 z^3 a^{-3} +7 z^3 a^{-5} +2 a^4 z^2-9 a^2 z^2-19 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -27 z^2+a^5 z-a^3 z-7 a z-11 z a^{-1} -8 z a^{-3} -2 z a^{-5} +3 a^2+5 a^{-2} + a^{-4} +8 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a316,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a36, K11a316,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a35"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ [math]\displaystyle{ t^4-5 t^3+14 t^2-25 t+31-25 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-4 q^5+8 q^4-13 q^3+17 q^2-19 q+20-16 q^{-1} +12 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5} }[/math] }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11a316,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a36, K11a316,}

Vassiliev invariants

V2 and V3:

(2, -1)

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 [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{28}{3} }[/math] [math]\displaystyle{ -\frac{52}{3} }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -\frac{176}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{224}{3} }[/math] [math]\displaystyle{ -\frac{416}{3} }[/math] [math]\displaystyle{ \frac{1231}{15} }[/math] [math]\displaystyle{ \frac{1516}{15} }[/math] [math]\displaystyle{ -\frac{7916}{45} }[/math] [math]\displaystyle{ \frac{17}{9} }[/math] [math]\displaystyle{ -\frac{929}{15} }[/math]

V_2,1 through V_6,9 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of K11a35. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   5 1   4 7                 8 3     -5 5               9 5       4 3             10 8         -2 1           10 9           1 -1         7 11             4 -3       5 9               -4 -5     2 7                 5 -7   1 5                   -4 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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.

K11a34 K11a36