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

K11a124 K11a126

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,5,15,6 X20,8,21,7 X2,10,3,9 X16,11,17,12 X18,14,19,13 X8,15,9,16 X22,17,1,18 X6,20,7,19 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 20 2 16 18 8 22 6 12

Alexander Polynomial:

- t-4 + 6t-3 - 19t-2 + 38t-1 - 47 + 38t - 19t2 + 6t3 - t4

Conway Polynomial:

1 - 3z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{175, 2}

Jones Polynomial:

q-3 - 4q-2 + 10q-1 - 17 + 24q - 28q2 + 29q3 - 25q4 + 19q5 - 12q6 + 5q7 - q8

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

{K11a297, ...}

A2 (sl(3)) Invariant:

q-8 - 2q-6 + 4q-4 - 2q-2 - 1 + 5q2 - 6q4 + 5q6 - 3q8 + q10 + 3q12 - 4q14 + 5q16 - 3q18 - q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6 - a-6z2 - a-6z4 + 3a-4 + 7a-4z2 + 6a-4z4 + 2a-4z6 - 3a-2 - 10a-2z2 - 11a-2z4 - 5a-2z6 - a-2z8 + 2 + 4z2 + 3z4 + z6

Kauffman Polynomial:

a-9z5 - 3a-8z4 + 5a-8z6 - a-7z + 5a-7z3 - 15a-7z5 + 12a-7z7 + a-6 - 4a-6z2 + 12a-6z4 - 24a-6z6 + 16a-6z8 - 2a-5z + 14a-5z3 - 22a-5z5 - 2a-5z7 + 11a-5z9 + 3a-4 - 16a-4z2 + 49a-4z4 - 66a-4z6 + 26a-4z8 + 3a-4z10 - 2a-3z + 12a-3z3 - 3a-3z5 - 27a-3z7 + 19a-3z9 + 3a-2 - 20a-2z2 + 50a-2z4 - 55a-2z6 + 18a-2z8 + 3a-2z10 - 2a-1z + 8a-1z3 - 5a-1z5 - 9a-1z7 + 8a-1z9 + 2 - 7z2 + 14z4 - 17z6 + 8z8 - az + 5az3 - 8az5 + 4az7 + a2z2 - 2a2z4 + a2z6

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

{0, 1}

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 11125. 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 r = 6 r = 7 j = 17                       1 j = 15                     4   j = 13                   8 1   j = 11                 11 4     j = 9               14 8       j = 7             15 11         j = 5           13 14           j = 3         11 15             j = 1       7 14               j = -1     3 10                 j = -3   1 7                   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, Alternating, 125]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 125]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 5, 15, 6], X[20, 8, 21, 7], > X[2, 10, 3, 9], X[16, 11, 17, 12], X[18, 14, 19, 13], X[8, 15, 9, 16], > X[22, 17, 1, 18], X[6, 20, 7, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 125]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -11, 7, -3, 8, -6, 9, -7, 10, > -4, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 125]]
Out[5]=	DTCode[4, 10, 14, 20, 2, 16, 18, 8, 22, 6, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 125]][t]
Out[6]=	-4 6 19 38 2 3 4 -47 - t + -- - -- + -- + 38 t - 19 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 125]][z]
Out[7]=	4 6 8 1 - 3 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 125]}
In[9]:=	{KnotDet[Knot[11, Alternating, 125]], KnotSignature[Knot[11, Alternating, 125]]}
Out[9]=	{175, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 125]][q]
Out[10]=	-3 4 10 2 3 4 5 6 7 8 -17 + q - -- + -- + 24 q - 28 q + 29 q - 25 q + 19 q - 12 q + 5 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 125], Knot[11, Alternating, 297]}
In[12]:=	A2Invariant[Knot[11, Alternating, 125]][q]
Out[12]=	-8 2 4 2 2 4 6 8 10 12 14 -1 + q - -- + -- - -- + 5 q - 6 q + 5 q - 3 q + q + 3 q - 4 q + 6 4 2 q q q 16 18 20 22 24 > 5 q - 3 q - q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 125]][a, z]
Out[13]=	2 2 2 4 4 4 -6 3 3 2 z 7 z 10 z 4 z 6 z 11 z 6 2 - a + -- - -- + 4 z - -- + ---- - ----- + 3 z - -- + ---- - ----- + z + 4 2 6 4 2 6 4 2 a a a a a a a a 6 6 8 2 z 5 z z > ---- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 125]][a, z]
Out[14]=	2 2 2 -6 3 3 z 2 z 2 z 2 z 2 4 z 16 z 20 z 2 + a + -- + -- - -- - --- - --- - --- - a z - 7 z - ---- - ----- - ----- + 4 2 7 5 3 a 6 4 2 a a a a a a a a 3 3 3 3 4 4 2 2 5 z 14 z 12 z 8 z 3 4 3 z 12 z > a z + ---- + ----- + ----- + ---- + 5 a z + 14 z - ---- + ----- + 7 5 3 a 8 6 a a a a a 4 4 5 5 5 5 5 49 z 50 z 2 4 z 15 z 22 z 3 z 5 z 5 > ----- + ----- - 2 a z + -- - ----- - ----- - ---- - ---- - 8 a z - 4 2 9 7 5 3 a a a a a a a 6 6 6 6 7 7 7 6 5 z 24 z 66 z 55 z 2 6 12 z 2 z 27 z > 17 z + ---- - ----- - ----- - ----- + a z + ----- - ---- - ----- - 8 6 4 2 7 5 3 a a a a a a a 7 8 8 8 9 9 9 9 z 7 8 16 z 26 z 18 z 11 z 19 z 8 z > ---- + 4 a z + 8 z + ----- + ----- + ----- + ----- + ----- + ---- + a 6 4 2 5 3 a a a a a a 10 10 3 z 3 z > ----- + ----- 4 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 125]], Vassiliev[3][Knot[11, Alternating, 125]]}
Out[15]=	{0, 1}
In[16]:=	Kh[Knot[11, Alternating, 125]][q, t]
Out[16]=	3 1 3 1 7 3 10 7 q 3 14 q + 11 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 15 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 > 13 q t + 14 q t + 15 q t + 11 q t + 14 q t + 8 q t + 11 q t + 11 5 13 5 13 6 15 6 17 7 > 4 q t + 8 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a125

K11a124 K11a126