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

K11n136 K11n138

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,3,13,4 X5,16,6,17 X7,14,8,15 X18,9,19,10 X20,11,21,12 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, -5, 10, -6, 11, -9}

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 7t-2 - 13t-1 + 15 - 13t + 7t2 - t3

Conway Polynomial:

1 + 6z2 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{K11n109, ...}

Determinant and Signature:

{57, -4}

Jones Polynomial:

- 2q-9 + 4q-8 - 7q-7 + 9q-6 - 9q-5 + 10q-4 - 7q-3 + 5q-2 - 3q-1 + 1

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

{...}

A2 (sl(3)) Invariant:

- 2q-28 - q-26 - 2q-22 + 2q-20 + 2q-16 + 3q-14 + 3q-10 - 2q-8 - q-2 + 1

HOMFLY-PT Polynomial:

2a2z2 + a2z4 - 2a4z2 - 3a4z4 - a4z6 + 4a6 + 8a6z2 + 3a6z4 - 3a8 - 2a8z2

Kauffman Polynomial:

2a2z2 - 3a2z4 + a2z6 + 8a3z3 - 10a3z5 + 3a3z7 - 2a4z2 + 5a4z4 - 8a4z6 + 3a4z8 + a5z + 5a5z3 - 12a5z5 + 3a5z7 + a5z9 - 4a6 + 3a6z2 + 3a6z4 - 10a6z6 + 5a6z8 + 5a7z - 6a7z3 - 2a7z5 + 2a7z7 + a7z9 - 3a8 + 6a8z2 - 3a8z4 + 2a8z8 + a9z + 2a9z7 - a10z2 + 2a10z4 + a10z6 - 3a11z + 3a11z3

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

{6, -14}

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=-4 is the signature of 11137. 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 = 1                   1 j = -1                 2   j = -3               3 1   j = -5             5 3     j = -7           5 2       j = -9         4 5         j = -11       5 5           j = -13     2 4             j = -15   2 5               j = -17   2                 j = -19 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, 137]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 137]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[5, 16, 6, 17], X[7, 14, 8, 15], > X[18, 9, 19, 10], X[20, 11, 21, 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, 137]]
Out[4]=	GaussCode[1, -7, 2, -1, -3, 8, -4, -11, 5, -10, 6, -2, 7, 4, -8, 3, 9, -5, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 137]]
Out[5]=	DTCode[4, 12, -16, -14, 18, 20, 2, -6, 22, 10, 8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 137]][t]
Out[6]=	-3 7 13 2 3 15 - t + -- - -- - 13 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 137]][z]
Out[7]=	2 4 6 1 + 6 z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 109], Knot[11, NonAlternating, 137]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 137]], KnotSignature[Knot[11, NonAlternating, 137]]}
Out[9]=	{57, -4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 137]][q]
Out[10]=	2 4 7 9 9 10 7 5 3 1 - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q 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, 137]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 137]][q]
Out[12]=	2 -26 2 2 2 3 3 2 -2 1 - --- - q - --- + --- + --- + --- + --- - -- - q 28 22 20 16 14 10 8 q q q q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 137]][a, z]
Out[13]=	6 8 2 2 4 2 6 2 8 2 2 4 4 4 4 a - 3 a + 2 a z - 2 a z + 8 a z - 2 a z + a z - 3 a z + 6 4 4 6 > 3 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 137]][a, z]
Out[14]=	6 8 5 7 9 11 2 2 4 2 6 2 -4 a - 3 a + a z + 5 a z + a z - 3 a z + 2 a z - 2 a z + 3 a z + 8 2 10 2 3 3 5 3 7 3 11 3 2 4 > 6 a z - a z + 8 a z + 5 a z - 6 a z + 3 a z - 3 a z + 4 4 6 4 8 4 10 4 3 5 5 5 7 5 > 5 a z + 3 a z - 3 a z + 2 a z - 10 a z - 12 a z - 2 a z + 2 6 4 6 6 6 10 6 3 7 5 7 7 7 > a z - 8 a z - 10 a z + a z + 3 a z + 3 a z + 2 a z + 9 7 4 8 6 8 8 8 5 9 7 9 > 2 a z + 3 a z + 5 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 137]], Vassiliev[3][Knot[11, NonAlternating, 137]]}
Out[15]=	{6, -14}
In[16]:=	Kh[Knot[11, NonAlternating, 137]][q, t]
Out[16]=	3 3 2 2 2 5 2 4 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q q t q t q t q t q t q t q t 5 4 5 5 2 5 t 2 t 2 > ------ + ----- + ----- + ----- + ---- + ---- + -- + --- + q t 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n137

K11n136 K11n138