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

K11n147 K11n149

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X14,4,15,3 X5,19,6,18 X7,20,8,21 X9,1,10,22 X11,6,12,7 X2,14,3,13 X15,9,16,8 X17,10,18,11 X19,13,20,12 X21,16,22,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-4 + 5t-3 - 10t-2 + 14t-1 - 15 + 14t - 10t2 + 5t3 - t4

Conway Polynomial:

1 + 3z2 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a223, ...}

Determinant and Signature:

{75, 2}

Jones Polynomial:

q-3 - 4q-2 + 7q-1 - 10 + 13q - 12q2 + 12q3 - 9q4 + 5q5 - 2q6

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

{K11n168, ...}

A2 (sl(3)) Invariant:

q-8 - 2q-6 + q-4 - 2q-2 + 2q2 - q4 + 6q6 - q8 + 3q10 - q12 - 2q14 + q16 - 2q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 4a-4z2 + 4a-4z4 + a-4z6 + 3a-2 - a-2z2 - 7a-2z4 - 5a-2z6 - a-2z8 - 1 + z2 + 3z4 + z6

Kauffman Polynomial:

- 3a-7z + 3a-7z3 + a-6 - 4a-6z2 + 4a-6z4 + a-6z6 - 4a-5z + 9a-5z3 - 6a-5z5 + 4a-5z7 - 8a-4z2 + 21a-4z4 - 16a-4z6 + 6a-4z8 - 2a-3z + 12a-3z3 - 10a-3z5 - a-3z7 + 3a-3z9 - 3a-2 - 8a-2z2 + 33a-2z4 - 36a-2z6 + 12a-2z8 - a-1z + 12a-1z3 - 15a-1z5 - a-1z7 + 3a-1z9 - 1 - 4z2 + 14z4 - 18z6 + 6z8 + 6az3 - 11az5 + 4az7 - 2a2z4 + a2z6

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

{3, 4}

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 11148. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

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, 148]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 148]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[5, 19, 6, 18], X[7, 20, 8, 21], > X[9, 1, 10, 22], X[11, 6, 12, 7], X[2, 14, 3, 13], X[15, 9, 16, 8], > X[17, 10, 18, 11], X[19, 13, 20, 12], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 148]]
Out[4]=	GaussCode[1, -7, 2, -1, -3, 6, -4, 8, -5, 9, -6, 10, 7, -2, -8, 11, -9, 3, -10, > 4, -11, 5]
In[5]:=	DTCode[Knot[11, NonAlternating, 148]]
Out[5]=	DTCode[4, 14, -18, -20, -22, -6, 2, -8, -10, -12, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 148]][t]
Out[6]=	-4 5 10 14 2 3 4 -15 - t + -- - -- + -- + 14 t - 10 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 148]][z]
Out[7]=	2 6 8 1 + 3 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 223], Knot[11, NonAlternating, 148]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 148]], KnotSignature[Knot[11, NonAlternating, 148]]}
Out[9]=	{75, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 148]][q]
Out[10]=	-3 4 7 2 3 4 5 6 -10 + q - -- + - + 13 q - 12 q + 12 q - 9 q + 5 q - 2 q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 148], Knot[11, NonAlternating, 168]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 148]][q]
Out[12]=	-8 2 -4 2 2 4 6 8 10 12 14 16 q - -- + q - -- + 2 q - q + 6 q - q + 3 q - q - 2 q + q - 6 2 q q 18 20 22 > 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 148]][a, z]
Out[13]=	2 2 2 4 4 6 6 8 -6 3 2 z 4 z z 4 4 z 7 z 6 z 5 z z -1 - a + -- + z - -- + ---- - -- + 3 z + ---- - ---- + z + -- - ---- - -- 2 6 4 2 4 2 4 2 2 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 148]][a, z]
Out[14]=	2 2 2 3 3 -6 3 3 z 4 z 2 z z 2 4 z 8 z 8 z 3 z 9 z -1 + a - -- - --- - --- - --- - - - 4 z - ---- - ---- - ---- + ---- + ---- + 2 7 5 3 a 6 4 2 7 5 a a a a a a a a a 3 3 4 4 4 5 12 z 12 z 3 4 4 z 21 z 33 z 2 4 6 z > ----- + ----- + 6 a z + 14 z + ---- + ----- + ----- - 2 a z - ---- - 3 a 6 4 2 5 a a a a a 5 5 6 6 6 7 7 10 z 15 z 5 6 z 16 z 36 z 2 6 4 z z > ----- - ----- - 11 a z - 18 z + -- - ----- - ----- + a z + ---- - -- - 3 a 6 4 2 5 3 a a a a a a 7 8 8 9 9 z 7 8 6 z 12 z 3 z 3 z > -- + 4 a z + 6 z + ---- + ----- + ---- + ---- a 4 2 3 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 148]], Vassiliev[3][Knot[11, NonAlternating, 148]]}
Out[15]=	{3, 4}
In[16]:=	Kh[Knot[11, NonAlternating, 148]][q, t]
Out[16]=	3 1 3 1 4 3 6 4 q 3 7 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 6 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 6 q t + 6 q t + 6 q t + 3 q t + 6 q t + 2 q t + 3 q t + 13 5 > 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n148

K11n147 K11n149