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

K11a56 K11a58

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X16,6,17,5 X2837 X20,9,21,10 X22,11,1,12 X18,13,19,14 X6,16,7,15 X12,17,13,18 X14,19,15,20 X10,21,11,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 16 2 20 22 18 6 12 14 10

Alexander Polynomial:

- t-4 + 5t-3 - 12t-2 + 20t-1 - 23 + 20t - 12t2 + 5t3 - t4

Conway Polynomial:

1 + z2 - 2z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a108, K11a139, K11a231, ...}

Determinant and Signature:

{99, -2}

Jones Polynomial:

q-7 - 3q-6 + 7q-5 - 12q-4 + 14q-3 - 16q-2 + 16q-1 - 12 + 10q - 5q2 + 2q3 - q4

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

{K11a231, ...}

A2 (sl(3)) Invariant:

q-20 - q-18 + 3q-16 - q-14 - q-12 - 6q-8 + q-6 - 3q-4 + 4q-2 + 5 + 2q2 + 4q4 - 2q6 - q8 - q10 - q12

HOMFLY-PT Polynomial:

- 4a-2 - 4a-2z2 - a-2z4 + 12 + 18z2 + 10z4 + 2z6 - 10a2 - 19a2z2 - 15a2z4 - 6a2z6 - a2z8 + 3a4 + 6a4z2 + 4a4z4 + a4z6

Kauffman Polynomial:

- 4a-3z + 8a-3z3 - 5a-3z5 + a-3z7 + 4a-2 - 8a-2z2 + 11a-2z4 - 8a-2z6 + 2a-2z8 - 11a-1z + 20a-1z3 - 9a-1z5 - 3a-1z7 + 2a-1z9 + 12 - 24z2 + 28z4 - 21z6 + 4z8 + z10 - 17az + 29az3 - 14az5 - 9az7 + 6az9 + 10a2 - 22a2z2 + 30a2z4 - 31a2z6 + 10a2z8 + a2z10 - 16a3z + 35a3z3 - 29a3z5 + 4a3z7 + 4a3z9 + 3a4 - 2a4z2 + 6a4z4 - 12a4z6 + 8a4z8 - 6a5z + 16a5z3 - 16a5z5 + 9a5z7 + 3a6z2 - 6a6z4 + 6a6z6 - 2a7z3 + 3a7z5 - a8z2 + a8z4

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

{1, 3}

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 1157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                       1 j = 7                     1   j = 5                   4 1   j = 3                 6 1     j = 1               6 4       j = -1             10 6         j = -3           7 7           j = -5         7 9             j = -7       5 7               j = -9     2 7                 j = -11   1 5                   j = -13   2                     j = -15 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, Alternating, 57]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 57]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[16, 6, 17, 5], X[2, 8, 3, 7], > X[20, 9, 21, 10], X[22, 11, 1, 12], X[18, 13, 19, 14], X[6, 16, 7, 15], > X[12, 17, 13, 18], X[14, 19, 15, 20], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 57]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -9, 7, -10, 8, -3, 9, -7, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 57]]
Out[5]=	DTCode[4, 8, 16, 2, 20, 22, 18, 6, 12, 14, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 57]][t]
Out[6]=	-4 5 12 20 2 3 4 -23 - t + -- - -- + -- + 20 t - 12 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 57]][z]
Out[7]=	2 4 6 8 1 + z - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 57], Knot[11, Alternating, 108], > Knot[11, Alternating, 139], Knot[11, Alternating, 231]}
In[9]:=	{KnotDet[Knot[11, Alternating, 57]], KnotSignature[Knot[11, Alternating, 57]]}
Out[9]=	{99, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 57]][q]
Out[10]=	-7 3 7 12 14 16 16 2 3 4 -12 + q - -- + -- - -- + -- - -- + -- + 10 q - 5 q + 2 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 57], Knot[11, Alternating, 231]}
In[12]:=	A2Invariant[Knot[11, Alternating, 57]][q]
Out[12]=	-20 -18 3 -14 -12 6 -6 3 4 2 4 6 5 + q - q + --- - q - q - -- + q - -- + -- + 2 q + 4 q - 2 q - 16 8 4 2 q q q q 8 10 12 > q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 57]][a, z]
Out[13]=	2 4 4 2 4 2 4 z 2 2 4 2 4 z 12 - -- - 10 a + 3 a + 18 z - ---- - 19 a z + 6 a z + 10 z - -- - 2 2 2 a a a 2 4 4 4 6 2 6 4 6 2 8 > 15 a z + 4 a z + 2 z - 6 a z + a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 57]][a, z]
Out[14]=	4 2 4 4 z 11 z 3 5 2 12 + -- + 10 a + 3 a - --- - ---- - 17 a z - 16 a z - 6 a z - 24 z - 2 3 a a a 2 3 3 8 z 2 2 4 2 6 2 8 2 8 z 20 z 3 > ---- - 22 a z - 2 a z + 3 a z - a z + ---- + ----- + 29 a z + 2 3 a a a 4 3 3 5 3 7 3 4 11 z 2 4 4 4 > 35 a z + 16 a z - 2 a z + 28 z + ----- + 30 a z + 6 a z - 2 a 5 5 6 4 8 4 5 z 9 z 5 3 5 5 5 7 5 > 6 a z + a z - ---- - ---- - 14 a z - 29 a z - 16 a z + 3 a z - 3 a a 6 7 7 6 8 z 2 6 4 6 6 6 z 3 z 7 > 21 z - ---- - 31 a z - 12 a z + 6 a z + -- - ---- - 9 a z + 2 3 a a a 8 9 3 7 5 7 8 2 z 2 8 4 8 2 z 9 > 4 a z + 9 a z + 4 z + ---- + 10 a z + 8 a z + ---- + 6 a z + 2 a a 3 9 10 2 10 > 4 a z + z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 57]], Vassiliev[3][Knot[11, Alternating, 57]]}
Out[15]=	{1, 3}
In[16]:=	Kh[Knot[11, Alternating, 57]][q, t]
Out[16]=	7 10 1 2 1 5 2 7 5 7 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 7 9 7 6 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 6 q t + 4 q t + 6 q t + q t + 4 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a57

K11a56 K11a58