10 13

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10 13 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,18,6,19 X13,1,14,20 X19,15,20,14 X7,16,8,17 X15,8,16,9 X17,6,18,7
Gauss code	-1, 4, -3, 1, -5, 10, -8, 9, -2, 3, -4, 2, -6, 7, -9, 8, -10, 5, -7, 6
Dowker-Thistlethwaite code	4 10 18 16 12 2 20 8 6 14
Conway Notation	[4222]

Minimum Braid Representative: Length is 11, width is 6. Braid index is 6.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-5] Hyperbolic Volume 10.5785 A-Polynomial See Data:10 13/A-polynomial

[edit Notes for 10 13's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 10 13's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-13t+23-13t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}-5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 53, 0 }
Jones polynomial	$q^{4}-2q^{3}+5q^{2}-7q+8-9q^{-1}+8q^{-2}-6q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-2z^{2}a^{4}-a^{4}+z^{4}a^{2}+a^{2}+z^{4}-z^{2}-1-2z^{2}a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$a^{3}z^{9}+az^{9}+2a^{4}z^{8}+4a^{2}z^{8}+2z^{8}+2a^{5}z^{7}+az^{7}+3z^{7}a^{-1}+a^{6}z^{6}-5a^{4}z^{6}-9a^{2}z^{6}+3z^{6}a^{-2}-7a^{5}z^{5}-4a^{3}z^{5}-2az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}-4a^{6}z^{4}+a^{4}z^{4}+6a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-3z^{4}+6a^{5}z^{3}+a^{3}z^{3}+3z^{3}a^{-1}-2z^{3}a^{-3}+4a^{6}z^{2}+2a^{4}z^{2}-a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+4z^{2}-a^{5}z+a^{3}z-2za^{-1}-a^{6}-a^{4}-a^{2}+a^{-4}-1$
The A2 invariant	$q^{20}+q^{18}-q^{16}+q^{14}-2q^{10}+2q^{8}-2+q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-q^{-10}+q^{-12}+q^{-14}$
The G2 invariant	Data:10 13/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_13.

A1 Invariants.

Weight	Invariant
1	$q^{13}-q^{11}+2q^{9}-2q^{7}+2q^{5}-q^{3}-q+q^{-1}-2q^{-3}+3q^{-5}-q^{-7}+q^{-9}$
2	$q^{38}-q^{36}-q^{34}+4q^{32}-2q^{30}-6q^{28}+8q^{26}+2q^{24}-11q^{22}+7q^{20}+6q^{18}-13q^{16}+3q^{14}+9q^{12}-8q^{10}-2q^{8}+7q^{6}+2q^{4}-6q^{2}+11q^{-2}-7q^{-4}-7q^{-6}+13q^{-8}-5q^{-10}-8q^{-12}+9q^{-14}-2q^{-16}-4q^{-18}+4q^{-20}-q^{-24}+q^{-26}$
3	$q^{75}-q^{73}-q^{71}+q^{69}+3q^{67}-2q^{65}-7q^{63}+q^{61}+13q^{59}+3q^{57}-17q^{55}-12q^{53}+20q^{51}+23q^{49}-17q^{47}-34q^{45}+6q^{43}+41q^{41}+6q^{39}-42q^{37}-19q^{35}+39q^{33}+31q^{31}-31q^{29}-37q^{27}+23q^{25}+40q^{23}-13q^{21}-42q^{19}+7q^{17}+37q^{15}+5q^{13}-32q^{11}-16q^{9}+22q^{7}+26q^{5}-10q^{3}-36q-5q^{-1}+40q^{-3}+22q^{-5}-40q^{-7}-29q^{-9}+32q^{-11}+36q^{-13}-24q^{-15}-30q^{-17}+14q^{-19}+26q^{-21}-9q^{-23}-18q^{-25}+7q^{-27}+9q^{-29}-4q^{-31}-7q^{-33}+4q^{-35}+4q^{-37}-3q^{-39}-3q^{-41}+2q^{-43}+q^{-45}-q^{-49}+q^{-51}$

A2 Invariants.

Weight	Invariant
1,0	$q^{20}+q^{18}-q^{16}+q^{14}-2q^{10}+2q^{8}-2+q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-q^{-10}+q^{-12}+q^{-14}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 13"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$2t^{2}-13t+23-13t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{4}-5z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 53, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{4}-2q^{3}+5q^{2}-7q+8-9q^{-1}+8q^{-2}-6q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$a^{6}-2z^{2}a^{4}-a^{4}+z^{4}a^{2}+a^{2}+z^{4}-z^{2}-1-2z^{2}a^{-2}+a^{-4}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{3}z^{9}+az^{9}+2a^{4}z^{8}+4a^{2}z^{8}+2z^{8}+2a^{5}z^{7}+az^{7}+3z^{7}a^{-1}+a^{6}z^{6}-5a^{4}z^{6}-9a^{2}z^{6}+3z^{6}a^{-2}-7a^{5}z^{5}-4a^{3}z^{5}-2az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}-4a^{6}z^{4}+a^{4}z^{4}+6a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-3z^{4}+6a^{5}z^{3}+a^{3}z^{3}+3z^{3}a^{-1}-2z^{3}a^{-3}+4a^{6}z^{2}+2a^{4}z^{2}-a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+4z^{2}-a^{5}z+a^{3}z-2za^{-1}-a^{6}-a^{4}-a^{2}+a^{-4}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {...}

Vassiliev invariants

V2 and V3:

(-5, 2)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-20$ $16$ $200$ ${\frac {698}{3}}$ ${\frac {262}{3}}$ $-320$ $-{\frac {1376}{3}}$ $-{\frac {320}{3}}$ $-80$ $-{\frac {4000}{3}}$ $128$ $-{\frac {13960}{3}}$ $-{\frac {5240}{3}}$ $-{\frac {21823}{6}}$ ${\frac {1582}{3}}$ $-{\frac {27422}{9}}$ ${\frac {7067}{18}}$ $-{\frac {4063}{6}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 10 13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                     1 1 7                   1   -1 5                 4 1   3 3               3 1     -2 1             5 4       1 -1           5 4         -1 -3         3 4           -1 -5       3 5             2 -7     1 3               -2 -9   1 3                 2 -11   1                   -1 -13 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{12}-2q^{11}+q^{10}+5q^{9}-10q^{8}+3q^{7}+16q^{6}-27q^{5}+6q^{4}+34q^{3}-47q^{2}+6q+52-58q^{-1}+60q^{-3}-53q^{-4}-9q^{-5}+54q^{-6}-36q^{-7}-15q^{-8}+38q^{-9}-17q^{-10}-14q^{-11}+20q^{-12}-4q^{-13}-8q^{-14}+6q^{-15}-2q^{-17}+q^{-18}$
3	$q^{24}-2q^{23}+q^{22}+q^{21}+2q^{20}-7q^{19}+q^{18}+8q^{17}+2q^{16}-18q^{15}+4q^{14}+21q^{13}-43q^{11}+13q^{10}+56q^{9}-12q^{8}-87q^{7}+19q^{6}+116q^{5}-16q^{4}-148q^{3}+8q^{2}+178q+2-193q^{-1}-23q^{-2}+204q^{-3}+38q^{-4}-197q^{-5}-61q^{-6}+188q^{-7}+75q^{-8}-165q^{-9}-91q^{-10}+139q^{-11}+104q^{-12}-112q^{-13}-108q^{-14}+79q^{-15}+110q^{-16}-50q^{-17}-100q^{-18}+21q^{-19}+87q^{-20}-2q^{-21}-65q^{-22}-14q^{-23}+47q^{-24}+15q^{-25}-25q^{-26}-17q^{-27}+15q^{-28}+10q^{-29}-5q^{-30}-7q^{-31}+3q^{-32}+2q^{-33}-2q^{-35}+q^{-36}$
4	$q^{40}-2q^{39}+q^{38}+q^{37}-2q^{36}+5q^{35}-9q^{34}+3q^{33}+6q^{32}-7q^{31}+16q^{30}-25q^{29}+7q^{28}+13q^{27}-23q^{26}+40q^{25}-42q^{24}+25q^{23}+17q^{22}-73q^{21}+61q^{20}-56q^{19}+96q^{18}+50q^{17}-179q^{16}+21q^{15}-92q^{14}+251q^{13}+170q^{12}-303q^{11}-114q^{10}-214q^{9}+435q^{8}+399q^{7}-351q^{6}-286q^{5}-434q^{4}+541q^{3}+644q^{2}-283q-374-665q^{-1}+517q^{-2}+782q^{-3}-156q^{-4}-334q^{-5}-803q^{-6}+401q^{-7}+773q^{-8}-29q^{-9}-206q^{-10}-834q^{-11}+240q^{-12}+664q^{-13}+87q^{-14}-42q^{-15}-781q^{-16}+53q^{-17}+484q^{-18}+189q^{-19}+145q^{-20}-653q^{-21}-127q^{-22}+253q^{-23}+228q^{-24}+310q^{-25}-440q^{-26}-222q^{-27}+18q^{-28}+160q^{-29}+377q^{-30}-196q^{-31}-187q^{-32}-125q^{-33}+30q^{-34}+306q^{-35}-22q^{-36}-74q^{-37}-134q^{-38}-59q^{-39}+165q^{-40}+30q^{-41}+9q^{-42}-68q^{-43}-64q^{-44}+57q^{-45}+15q^{-46}+24q^{-47}-17q^{-48}-31q^{-49}+15q^{-50}+10q^{-52}-q^{-53}-9q^{-54}+4q^{-55}-q^{-56}+2q^{-57}-2q^{-59}+q^{-60}$
5	$q^{60}-2q^{59}+q^{58}+q^{57}-2q^{56}+q^{55}+3q^{54}-7q^{53}+q^{52}+7q^{51}-4q^{50}+2q^{49}+7q^{48}-18q^{47}-4q^{46}+11q^{45}+2q^{44}+11q^{43}+20q^{42}-24q^{41}-31q^{40}-12q^{39}-8q^{38}+45q^{37}+80q^{36}+13q^{35}-61q^{34}-115q^{33}-112q^{32}+55q^{31}+236q^{30}+201q^{29}-q^{28}-291q^{27}-450q^{26}-128q^{25}+426q^{24}+676q^{23}+369q^{22}-409q^{21}-1059q^{20}-748q^{19}+402q^{18}+1378q^{17}+1226q^{16}-177q^{15}-1719q^{14}-1811q^{13}-121q^{12}+1936q^{11}+2388q^{10}+568q^{9}-2021q^{8}-2925q^{7}-1073q^{6}+1978q^{5}+3332q^{4}+1559q^{3}-1786q^{2}-3585q-2009+1546q^{-1}+3683q^{-2}+2315q^{-3}-1236q^{-4}-3642q^{-5}-2550q^{-6}+980q^{-7}+3493q^{-8}+2629q^{-9}-682q^{-10}-3283q^{-11}-2679q^{-12}+453q^{-13}+3017q^{-14}+2624q^{-15}-173q^{-16}-2696q^{-17}-2585q^{-18}-85q^{-19}+2338q^{-20}+2471q^{-21}+374q^{-22}-1902q^{-23}-2326q^{-24}-672q^{-25}+1431q^{-26}+2109q^{-27}+921q^{-28}-909q^{-29}-1790q^{-30}-1127q^{-31}+382q^{-32}+1420q^{-33}+1197q^{-34}+89q^{-35}-948q^{-36}-1161q^{-37}-475q^{-38}+491q^{-39}+975q^{-40}+710q^{-41}-46q^{-42}-708q^{-43}-788q^{-44}-285q^{-45}+374q^{-46}+722q^{-47}+505q^{-48}-91q^{-49}-542q^{-50}-550q^{-51}-165q^{-52}+328q^{-53}+518q^{-54}+270q^{-55}-128q^{-56}-366q^{-57}-320q^{-58}-23q^{-59}+246q^{-60}+258q^{-61}+93q^{-62}-104q^{-63}-195q^{-64}-110q^{-65}+37q^{-66}+106q^{-67}+90q^{-68}+15q^{-69}-63q^{-70}-57q^{-71}-12q^{-72}+14q^{-73}+32q^{-74}+21q^{-75}-11q^{-76}-17q^{-77}-q^{-78}-3q^{-79}+3q^{-80}+9q^{-81}-2q^{-82}-5q^{-83}+2q^{-84}-q^{-86}+2q^{-87}-2q^{-89}+q^{-90}$
6	Not Available
7	Not Available

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 13]]
Out[2]=	PD[X[1, 4, 2, 5], X[9, 12, 10, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[5, 18, 6, 19], X[13, 1, 14, 20], X[19, 15, 20, 14], X[7, 16, 8, 17], X[15, 8, 16, 9], X[17, 6, 18, 7]]
In[3]:=	GaussCode[Knot[10, 13]]
Out[3]=	GaussCode[-1, 4, -3, 1, -5, 10, -8, 9, -2, 3, -4, 2, -6, 7, -9, 8, -10, 5, -7, 6]
In[4]:=	DTCode[Knot[10, 13]]
Out[4]=	DTCode[4, 10, 18, 16, 12, 2, 20, 8, 6, 14]
In[5]:=	br = BR[Knot[10, 13]]
Out[5]=	BR[6, {-1, -1, -2, 1, 3, -2, -4, 3, 5, -4, 5}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{6, 11}
In[7]:=	BraidIndex[Knot[10, 13]]
Out[7]=	6
In[8]:=	Show[DrawMorseLink[Knot[10, 13]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 13]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 13]][t]
Out[10]=	2 13 2 23 + -- - -- - 13 t + 2 t 2 t t
In[11]:=	Conway[Knot[10, 13]][z]
Out[11]=	2 4 1 - 5 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 13]}
In[13]:=	{KnotDet[Knot[10, 13]], KnotSignature[Knot[10, 13]]}
Out[13]=	{53, 0}
In[14]:=	Jones[Knot[10, 13]][q]
Out[14]=	-6 2 4 6 8 9 2 3 4 8 + q - -- + -- - -- + -- - - - 7 q + 5 q - 2 q + q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 13]}
In[16]:=	A2Invariant[Knot[10, 13]][q]
Out[16]=	-20 -18 -16 -14 2 2 2 4 6 8 -2 + q + q - q + q - --- + -- + q - 2 q + q + 2 q - 10 8 q q 10 12 14 q + q + q
In[17]:=	HOMFLYPT[Knot[10, 13]][a, z]
Out[17]=	2 -4 2 4 6 2 2 z 4 2 4 2 4 -1 + a + a - a + a - z - ---- - 2 a z + z + a z 2 a
In[18]:=	Kauffman[Knot[10, 13]][a, z]
Out[18]=	2 2 -4 2 4 6 2 z 3 5 2 2 z z -1 + a - a - a - a - --- + a z - a z + 4 z - ---- + -- - a 4 2 a a 3 3 2 2 4 2 6 2 2 z 3 z 3 3 5 3 4 a z + 2 a z + 4 a z - ---- + ---- + a z + 6 a z - 3 z + 3 a a 4 4 5 5 z 3 z 2 4 4 4 6 4 2 z 3 z 5 -- - ---- + 6 a z + a z - 4 a z + ---- - ---- - 2 a z - 4 2 3 a a a a 6 7 3 5 5 5 3 z 2 6 4 6 6 6 3 z 7 4 a z - 7 a z + ---- - 9 a z - 5 a z + a z + ---- + a z + 2 a a 5 7 8 2 8 4 8 9 3 9 2 a z + 2 z + 4 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 13]], Vassiliev[3][Knot[10, 13]]}
Out[19]=	{-5, 2}
In[20]:=	Kh[Knot[10, 13]][q, t]
Out[20]=	4 1 1 1 3 1 3 3 - + 5 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 5 3 4 5 3 3 2 5 2 ----- + ----- + ---- + --- + 4 q t + 3 q t + q t + 4 q t + 5 2 3 2 3 q t q t q t q t 5 3 7 3 9 4 q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 13], 2][q]
Out[21]=	-18 2 6 8 4 20 14 17 38 15 36 52 + q - --- + --- - --- - --- + --- - --- - --- + -- - -- - -- + 17 15 14 13 12 11 10 9 8 7 q q q q q q q q q q 54 9 53 60 58 2 3 4 5 6 -- - -- - -- + -- - -- + 6 q - 47 q + 34 q + 6 q - 27 q + 16 q + 6 5 4 3 q q q q q 7 8 9 10 11 12 3 q - 10 q + 5 q + q - 2 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

10_12 10_14