© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:

1037 1039

KnotPlot

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KnotPlot

PD Presentation:

X1425 X3,10,4,11 X5,12,6,13 X15,18,16,19 X7,17,8,16 X17,7,18,6 X13,20,14,1 X19,14,20,15 X11,8,12,9 X9,2,10,3

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 2 8 20 18 6 14

Minimum Braid Representative: Length is 12, width is 5 Braid index is 5

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 2 2 2 / NotAvailable 1

Alexander Polynomial:

- 4t-2 + 15t-1 - 21 + 15t - 4t2

Conway Polynomial:

1 - z2 - 4z4

Other knots with the same Alexander/Conway Polynomial:

{K11a166, ...}

Determinant and Signature:

{59, -2}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

q-28 - q-26 - q-24 + 2q-22 - q-20 + q-18 + q-16 - 2q-14 - 2q-10 + q-8 + q-6 - q-4 + 3q-2 + q4

HOMFLY-PT Polynomial:

1 + z2 + a2 - a2z4 - 2a4 - 3a4z2 - 2a4z4 + a6 - a6z4 + a8z2

Kauffman Polynomial:

1 - 2z2 + z4 - 2az3 + 2az5 - a2 + 2a2z2 - 3a2z4 + 3a2z6 + a3z3 - 2a3z5 + 3a3z7 - 2a4 + 8a4z2 - 8a4z4 + 2a4z6 + 2a4z8 + 3a5z3 - 7a5z5 + 3a5z7 + a5z9 - a6 + 2a6z2 + 3a6z4 - 10a6z6 + 5a6z8 - a7z + 8a7z3 - 13a7z5 + 3a7z7 + a7z9 + 4a8z4 - 8a8z6 + 3a8z8 - a9z + 8a9z3 - 10a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6

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

{-1, 2}

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 1038. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 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                   1   j = -1                 4 1   j = -3               4 2     j = -5             5 3       j = -7           5 4         j = -9         4 5           j = -11       3 5             j = -13     2 4               j = -15   1 3                 j = -17   2                   j = -19 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-26 - 3q-25 + 10q-23 - 11q-22 - 9q-21 + 29q-20 - 15q-19 - 29q-18 + 50q-17 - 10q-16 - 53q-15 + 63q-14 + q-13 - 71q-12 + 64q-11 + 13q-10 - 72q-9 + 51q-8 + 17q-7 - 53q-6 + 31q-5 + 11q-4 - 28q-3 + 15q-2 + 4q-1 - 10 + 5q + q2 - 2q3 + q4
3	q-51 - 3q-50 + 5q-48 + 5q-47 - 11q-46 - 15q-45 + 16q-44 + 31q-43 - 14q-42 - 53q-41 + 2q-40 + 78q-39 + 20q-38 - 98q-37 - 52q-36 + 107q-35 + 92q-34 - 107q-33 - 132q-32 + 95q-31 + 169q-30 - 72q-29 - 201q-28 + 44q-27 + 227q-26 - 13q-25 - 245q-24 - 21q-23 + 256q-22 + 50q-21 - 252q-20 - 80q-19 + 240q-18 + 99q-17 - 214q-16 - 106q-15 + 175q-14 + 105q-13 - 138q-12 - 87q-11 + 97q-10 + 70q-9 - 72q-8 - 40q-7 + 42q-6 + 30q-5 - 36q-4 - 8q-3 + 18q-2 + 8q-1 - 17 + q + 7q2 + 2q3 - 7q4 + 2q5 + q6 + q7 - 2q8 + q9
4	q-84 - 3q-83 + 5q-81 + 5q-79 - 18q-78 - 8q-77 + 17q-76 + 11q-75 + 40q-74 - 49q-73 - 54q-72 + 4q-71 + 22q-70 + 148q-69 - 38q-68 - 115q-67 - 89q-66 - 56q-65 + 299q-64 + 79q-63 - 81q-62 - 210q-61 - 289q-60 + 346q-59 + 231q-58 + 136q-57 - 202q-56 - 597q-55 + 199q-54 + 254q-53 + 452q-52 + 19q-51 - 804q-50 - 74q-49 + 69q-48 + 716q-47 + 378q-46 - 835q-45 - 342q-44 - 246q-43 + 855q-42 + 745q-41 - 735q-40 - 546q-39 - 576q-38 + 893q-37 + 1046q-36 - 564q-35 - 676q-34 - 864q-33 + 830q-32 + 1243q-31 - 327q-30 - 694q-29 - 1067q-28 + 633q-27 + 1253q-26 - 57q-25 - 528q-24 - 1092q-23 + 324q-22 + 1018q-21 + 136q-20 - 231q-19 - 883q-18 + 59q-17 + 627q-16 + 151q-15 + 23q-14 - 536q-13 - 47q-12 + 279q-11 + 55q-10 + 117q-9 - 241q-8 - 33q-7 + 91q-6 - 24q-5 + 93q-4 - 84q-3 + 27q-1 - 40 + 45q - 26q2 + 10q3 + 10q4 - 25q5 + 17q6 - 8q7 + 5q8 + 4q9 - 9q10 + 5q11 - 2q12 + q13 + q14 - 2q15 + q16

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]:=	PD[Knot[10, 38]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 12, 6, 13], X[15, 18, 16, 19], > X[7, 17, 8, 16], X[17, 7, 18, 6], X[13, 20, 14, 1], X[19, 14, 20, 15], > X[11, 8, 12, 9], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 38]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 6, -5, 9, -10, 2, -9, 3, -7, 8, -4, 5, -6, 4, -8, > 7]
In[4]:=	DTCode[Knot[10, 38]]
Out[4]=	DTCode[4, 10, 12, 16, 2, 8, 20, 18, 6, 14]
In[5]:=	br = BR[Knot[10, 38]]
Out[5]=	BR[5, {-1, -1, -1, -2, 1, -2, -2, -3, 2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 38]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 38]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 38]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 38]][t]
Out[10]=	4 15 2 -21 - -- + -- + 15 t - 4 t 2 t t
In[11]:=	Conway[Knot[10, 38]][z]
Out[11]=	2 4 1 - z - 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 38], Knot[11, Alternating, 166]}
In[13]:=	{KnotDet[Knot[10, 38]], KnotSignature[Knot[10, 38]]}
Out[13]=	{59, -2}
In[14]:=	Jones[Knot[10, 38]][q]
Out[14]=	-9 3 5 7 9 10 9 7 5 -2 + q - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 38]}
In[16]:=	A2Invariant[Knot[10, 38]][q]
Out[16]=	-28 -26 -24 2 -20 -18 -16 2 2 -8 -6 -4 q - q - q + --- - q + q + q - --- - --- + q + q - q + 22 14 10 q q q 3 4 > -- + q 2 q
In[17]:=	HOMFLYPT[Knot[10, 38]][a, z]
Out[17]=	2 4 6 2 4 2 8 2 2 4 4 4 6 4 1 + a - 2 a + a + z - 3 a z + a z - a z - 2 a z - a z
In[18]:=	Kauffman[Knot[10, 38]][a, z]
Out[18]=	2 4 6 7 9 2 2 2 4 2 6 2 1 - a - 2 a - a - a z - a z - 2 z + 2 a z + 8 a z + 2 a z + 10 2 3 3 3 5 3 7 3 9 3 4 2 4 > 2 a z - 2 a z + a z + 3 a z + 8 a z + 8 a z + z - 3 a z - 4 4 6 4 8 4 10 4 5 3 5 5 5 > 8 a z + 3 a z + 4 a z - 3 a z + 2 a z - 2 a z - 7 a z - 7 5 9 5 2 6 4 6 6 6 8 6 10 6 > 13 a z - 10 a z + 3 a z + 2 a z - 10 a z - 8 a z + a z + 3 7 5 7 7 7 9 7 4 8 6 8 8 8 > 3 a z + 3 a z + 3 a z + 3 a z + 2 a z + 5 a z + 3 a z + 5 9 7 9 > a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 38]], Vassiliev[3][Knot[10, 38]]}
Out[19]=	{-1, 2}
In[20]:=	Kh[Knot[10, 38]][q, t]
Out[20]=	2 4 1 2 1 3 2 4 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 5 4 5 5 4 5 3 4 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + q t + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 3 2 > q t
In[21]:=	ColouredJones[Knot[10, 38], 2][q]
Out[21]=	-26 3 10 11 9 29 15 29 50 10 53 63 -10 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- + 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q -13 71 64 13 72 51 17 53 31 11 28 15 4 > q - --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + -- + - + 5 q + 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 2 3 4 > q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1038

1037 1039