© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n139 K11n141

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,3,13,4 X5,16,6,17 X7,14,8,15 X20,9,21,10 X18,11,19,12 X2,13,3,14 X15,6,16,7 X22,18,1,17 X10,19,11,20 X8,21,9,22

Gauss Code:

{1, -7, 2, -1, -3, 8, -4, -11, 5, -10, 6, -2, 7, 4, -8, 3, 9, -6, 10, -5, 11, -9}

DT (Dowker-Thistlethwaite) Code:

4 12 -16 -14 20 18 2 -6 22 10 8

Alexander Polynomial:

- 2t-2 + 13t-1 - 21 + 13t - 2t2

Conway Polynomial:

1 + 5z2 - 2z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{51, -2}

Jones Polynomial:

- 2q-8 + 3q-7 - 5q-6 + 8q-5 - 8q-4 + 9q-3 - 7q-2 + 5q-1 - 3 + q

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- 2q-26 - 2q-24 + q-22 - q-20 + q-18 + 3q-16 + 2q-12 + q-8 + q-6 - 2q-4 + 2q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 - a2z4 + a4 + a4z2 - a4z4 + 2a6 + 3a6z2 - 2a8

Kauffman Polynomial:

- z2 + z4 - 4az3 + 3az5 + a2z2 - 5a2z4 + 4a2z6 + 4a3z3 - 6a3z5 + 4a3z7 + a4 - 5a4z2 + 9a4z4 - 6a4z6 + 3a4z8 + 2a5z + a5z3 - 3a5z5 + a5z7 + a5z9 - 2a6 - 6a6z2 + 15a6z4 - 12a6z6 + 4a6z8 + 9a7z - 17a7z3 + 9a7z5 - 3a7z7 + a7z9 - 2a8 + a8z2 - 2a8z6 + a8z8 + 7a9z - 10a9z3 + 3a9z5

V2 and V3, the type 2 and 3 Vassiliev invariants:

{5, -11}

Khovanov Homology: (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 11140. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                   1 j = 1                 2   j = -1               3 1   j = -3             5 3     j = -5           4 2       j = -7         4 5         j = -9       4 4           j = -11     1 4             j = -13   2 4               j = -15   1                 j = -17 2

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 140]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 140]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[5, 16, 6, 17], X[7, 14, 8, 15], > X[20, 9, 21, 10], X[18, 11, 19, 12], X[2, 13, 3, 14], X[15, 6, 16, 7], > X[22, 18, 1, 17], X[10, 19, 11, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 140]]
Out[4]=	GaussCode[1, -7, 2, -1, -3, 8, -4, -11, 5, -10, 6, -2, 7, 4, -8, 3, 9, -6, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 140]]
Out[5]=	DTCode[4, 12, -16, -14, 20, 18, 2, -6, 22, 10, 8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 140]][t]
Out[6]=	2 13 2 -21 - -- + -- + 13 t - 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 140]][z]
Out[7]=	2 4 1 + 5 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 140]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 140]], KnotSignature[Knot[11, NonAlternating, 140]]}
Out[9]=	{51, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 140]][q]
Out[10]=	2 3 5 8 8 9 7 5 -3 - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 140]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 140]][q]
Out[12]=	2 2 -22 -20 -18 3 2 -8 -6 2 2 2 4 -1 - --- - --- + q - q + q + --- + --- + q + q - -- + -- - q + q 26 24 16 12 4 2 q q q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 140]][a, z]
Out[13]=	4 6 8 2 4 2 6 2 2 4 4 4 a + 2 a - 2 a + z + a z + 3 a z - a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 140]][a, z]
Out[14]=	4 6 8 5 7 9 2 2 2 4 2 6 2 a - 2 a - 2 a + 2 a z + 9 a z + 7 a z - z + a z - 5 a z - 6 a z + 8 2 3 3 3 5 3 7 3 9 3 4 2 4 > a z - 4 a z + 4 a z + a z - 17 a z - 10 a z + z - 5 a z + 4 4 6 4 5 3 5 5 5 7 5 9 5 > 9 a z + 15 a z + 3 a z - 6 a z - 3 a z + 9 a z + 3 a z + 2 6 4 6 6 6 8 6 3 7 5 7 7 7 > 4 a z - 6 a z - 12 a z - 2 a z + 4 a z + a z - 3 a z + 4 8 6 8 8 8 5 9 7 9 > 3 a z + 4 a z + a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 140]], Vassiliev[3][Knot[11, NonAlternating, 140]]}
Out[15]=	{5, -11}
In[16]:=	Kh[Knot[11, NonAlternating, 140]][q, t]
Out[16]=	3 3 2 1 2 4 1 4 4 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 4 5 4 2 5 t 3 2 > ----- + ----- + ----- + ---- + ---- + - + 2 q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n140

K11n139 K11n141