4 1

3_1

5_1

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 4 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 4 1 at Knotilus!
	4_1 is also known as "the Figure Eight knot", as some people think it looks like a figure `8' in one of its common projections. See e.g. [1] . For two 4_1 knots along a closed loop, see 10_59, 10_60, K12a975, and K12a991.

Square depiction	Alternate square depiction	3D depiction	In "figure 8" form
A Neli-Kolam with 3x2 dot array[1]	In curved symmetrical form	Quasi-Celtic depiction	Symmetrical from parametric equation
Thurston's Trick [2]	Cylindrical depiction

Non-prime (compound) versions

• 

  Pseudo-Celtic ornamental knot pattern, with three figure-8 knots along a closed triangular loop.

• 

  (alternative)

• 

  Coat of arms surrounded by figure-8 knots

Knot presentations

Planar diagram presentation	X4251 X8615 X6374 X2738
Gauss code	1, -4, 3, -1, 2, -3, 4, -2
Dowker-Thistlethwaite code	4 6 8 2
Conway Notation	[22]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 4, width is 3, Braid index is 3		[{3, 5}, {6, 4}, {5, 2}, {1, 3}, {2, 6}, {4, 1}]

[edit Notes on presentations of 4 1]

Knot 4_1.

A graph, knot 4_1.

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["4 1"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X8615 X6374 X2738

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -1, 2, -3, 4, -2

In[6]:=

DTCode[K]

Out[6]=

4 6 8 2

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[22]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(3,\{-1,2,-1,2\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 3, 4, 3 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{3, 5}, {6, 4}, {5, 2}, {1, 3}, {2, 6}, {4, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index 3 Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-3] Hyperbolic Volume 2.02988 A-Polynomial See Data:4 1/A-polynomial

[edit Notes for 4 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(4,1))$ Rasmussen s-Invariant 0

[edit Notes for 4 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t-t^{-1}+3$
Conway polynomial	$1-z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, 0 }
Jones polynomial	$q^{2}+q^{-2}-q-q^{-1}+1$
HOMFLY-PT polynomial (db, data sources)	$a^{2}+a^{-2}-z^{2}-1$
Kauffman polynomial (db, data sources)	$a^{2}z^{2}+z^{2}a^{-2}-a^{2}-a^{-2}+az^{3}+z^{3}a^{-1}-az-za^{-1}+2z^{2}-1$
The A2 invariant	$q^{8}+q^{6}-1+q^{-6}+q^{-8}$
The G2 invariant	$q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^{4}-1-q^{-4}-q^{-10}+q^{-16}+q^{-18}+q^{-24}+q^{-26}+q^{-28}-q^{-30}+q^{-34}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 4_1.

A1 Invariants.

Weight	Invariant
1	$q^{5}+q^{-5}$
2	$q^{14}-q^{10}+q^{2}+1+q^{-2}-q^{-10}+q^{-14}$
3	$q^{27}-q^{23}-q^{21}+q^{17}+q^{11}+q^{9}+q^{-9}+q^{-11}+q^{-17}-q^{-21}-q^{-23}+q^{-27}$
4	$q^{44}-q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}-q^{26}+q^{24}+q^{22}-q^{18}-q^{16}+q^{4}+q^{2}+1+q^{-2}+q^{-4}-q^{-16}-q^{-18}+q^{-22}+q^{-24}-q^{-26}+q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}-q^{-40}+q^{-44}$
5	$q^{65}-q^{61}-q^{59}-q^{57}+q^{53}+2q^{51}+q^{49}-q^{45}-q^{43}+q^{39}+q^{37}-q^{35}-2q^{33}-q^{31}+q^{27}+q^{25}+q^{17}+q^{15}+q^{13}+q^{-13}+q^{-15}+q^{-17}+q^{-25}+q^{-27}-q^{-31}-2q^{-33}-q^{-35}+q^{-37}+q^{-39}-q^{-43}-q^{-45}+q^{-49}+2q^{-51}+q^{-53}-q^{-57}-q^{-59}-q^{-61}+q^{-65}$
6	$q^{90}-q^{86}-q^{84}-q^{82}+2q^{76}+2q^{74}+q^{72}-q^{68}-2q^{66}-2q^{64}+q^{62}+q^{60}+q^{58}-q^{54}-2q^{52}-2q^{50}+q^{48}+2q^{46}+2q^{44}+q^{42}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}-q^{26}-q^{24}-q^{22}+q^{6}+q^{4}+q^{2}+1+q^{-2}+q^{-4}+q^{-6}-q^{-22}-q^{-24}-q^{-26}+q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-42}+2q^{-44}+2q^{-46}+q^{-48}-2q^{-50}-2q^{-52}-q^{-54}+q^{-58}+q^{-60}+q^{-62}-2q^{-64}-2q^{-66}-q^{-68}+q^{-72}+2q^{-74}+2q^{-76}-q^{-82}-q^{-84}-q^{-86}+q^{-90}$

A2 Invariants.

Weight	Invariant
1,0	$q^{8}+q^{6}-1+q^{-6}+q^{-8}$
1,1	$q^{20}+2q^{16}-2q^{10}-2q^{8}+4q^{2}+2+4q^{-2}-2q^{-8}-2q^{-10}+2q^{-16}+q^{-20}$
2,0	$q^{20}+q^{18}+q^{16}-q^{14}-q^{12}-q^{10}-q^{8}+q^{4}+2q^{2}+2+2q^{-2}+q^{-4}-q^{-8}-q^{-10}-q^{-12}-q^{-14}+q^{-16}+q^{-18}+q^{-20}$
3,0	$q^{36}+q^{34}+q^{32}-2q^{28}-2q^{26}-2q^{24}+q^{18}+2q^{16}+3q^{14}+3q^{12}+2q^{10}+q^{8}-q^{4}-2q^{2}-2-2q^{-2}-q^{-4}+q^{-8}+2q^{-10}+3q^{-12}+3q^{-14}+2q^{-16}+q^{-18}-2q^{-24}-2q^{-26}-2q^{-28}+q^{-32}+q^{-34}+q^{-36}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{16}+q^{12}+q^{10}+q^{-10}+q^{-12}+q^{-16}$
1,0,0	$q^{11}+q^{9}+q^{7}-q-q^{-1}+q^{-7}+q^{-9}+q^{-11}$
1,0,1	$q^{26}+2q^{22}+2q^{20}+q^{18}+2q^{16}-2q^{14}-2q^{12}-4q^{10}-4q^{8}+2q^{4}+6q^{2}+7+6q^{-2}+2q^{-4}-4q^{-8}-4q^{-10}-2q^{-12}-2q^{-14}+2q^{-16}+q^{-18}+2q^{-20}+2q^{-22}+q^{-26}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-q^{10}-q^{8}+q^{2}+2+q^{-2}-q^{-8}-q^{-10}+q^{-14}+q^{-16}+q^{-18}+q^{-20}+q^{-22}$
1,0,0,0	$q^{14}+q^{12}+q^{10}+q^{8}-q^{2}-1-q^{-2}+q^{-8}+q^{-10}+q^{-12}+q^{-14}$

B2 Invariants.

Weight	Invariant
0,1	$q^{16}+q^{12}+q^{10}-2+q^{-10}+q^{-12}+q^{-16}$
1,0	$q^{26}+q^{18}+1+q^{-18}+q^{-26}$

B3 Invariants.

Weight	Invariant
1,0,0	$q^{38}+q^{30}+q^{26}+q^{22}-q^{2}+1-q^{-2}+q^{-22}+q^{-26}+q^{-30}+q^{-38}$

B4 Invariants.

Weight	Invariant
1,0,0,0	$q^{50}+q^{42}+q^{38}+q^{34}+q^{30}+q^{26}-q^{6}-q^{2}+1-q^{-2}-q^{-6}+q^{-26}+q^{-30}+q^{-34}+q^{-38}+q^{-42}+q^{-50}$

C3 Invariants.

Weight	Invariant
1,0,0	$q^{22}+q^{18}+q^{16}+q^{14}+q^{12}-q^{2}-2-q^{-2}+q^{-12}+q^{-14}+q^{-16}+q^{-18}+q^{-22}$

C4 Invariants.

Weight	Invariant
1,0,0,0	$q^{28}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-q^{4}-q^{2}-2-q^{-2}-q^{-4}+q^{-14}+q^{-16}+q^{-18}+q^{-20}+q^{-22}+q^{-24}+q^{-28}$

D4 Invariants.

Weight	Invariant
0,1,0,0	$q^{38}+q^{34}+3q^{32}+2q^{30}+q^{28}+4q^{26}+q^{24}-2q^{18}-5q^{16}-3q^{14}-3q^{12}-4q^{10}+3q^{6}+5q^{4}+6q^{2}+8+6q^{-2}+5q^{-4}+3q^{-6}-4q^{-10}-3q^{-12}-3q^{-14}-5q^{-16}-2q^{-18}+q^{-24}+4q^{-26}+q^{-28}+2q^{-30}+3q^{-32}+q^{-34}+q^{-38}$
1,0,0,0	$q^{22}+q^{18}+q^{16}+q^{14}+q^{12}-q^{2}-q^{-2}+q^{-12}+q^{-14}+q^{-16}+q^{-18}+q^{-22}$

G2 Invariants.

Weight	Invariant
1,0	$q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^{4}-1-q^{-4}-q^{-10}+q^{-16}+q^{-18}+q^{-24}+q^{-26}+q^{-28}-q^{-30}+q^{-34}+q^{-38}$

.

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["4 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t-t^{-1}+3$

In[5]:=

Conway[K][z]

Out[5]=

$1-z^{2}$

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]=

{ 5, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{2}+q^{-2}-q-q^{-1}+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{2}+z^{2}a^{-2}-a^{2}-a^{-2}+az^{3}+z^{3}a^{-1}-az-za^{-1}+2z^{2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n19,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["4 1"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ $-t-t^{-1}+3$, $q^{2}+q^{-2}-q-q^{-1}+1$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n19,}

Vassiliev invariants

V2 and V3:

(-1, 0)

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 $-4$ $0$ $8$ ${\frac {34}{3}}$ ${\frac {14}{3}}$ $0$ $0$ $0$ $0$ $-{\frac {32}{3}}$ $0$ $-{\frac {136}{3}}$ $-{\frac {56}{3}}$ $-{\frac {1231}{30}}$ ${\frac {142}{15}}$ $-{\frac {1742}{45}}$ ${\frac {79}{18}}$ $-{\frac {271}{30}}$

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

\   r    \    j   \ -2 -1 0 1 2 χ 5         1 1 3           0 1     1 1   0 -1   1 1     0 -3           0 -5 1         1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^6-q^5-q^4+2 q^3-q^2-q+3- q^{-1} - q^{-2} +2 q^{-3} - q^{-4} - q^{-5} + q^{-6} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-q^{11}-q^{10}+2 q^8-2 q^6+3 q^4-3 q^2+3-3 q^{-2} +3 q^{-4} -2 q^{-6} +2 q^{-8} - q^{-10} - q^{-11} + q^{-12} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-q^{19}-q^{18}+3 q^{15}-q^{14}-q^{13}-q^{12}-q^{11}+5 q^{10}-q^9-2 q^8-2 q^7-q^6+6 q^5-q^4-2 q^3-2 q^2-q+7- q^{-1} -2 q^{-2} -2 q^{-3} - q^{-4} +6 q^{-5} - q^{-6} -2 q^{-7} -2 q^{-8} - q^{-9} +5 q^{-10} - q^{-11} - q^{-12} - q^{-13} - q^{-14} +3 q^{-15} - q^{-18} - q^{-19} + q^{-20} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{30}-q^{29}-q^{28}+q^{25}+2 q^{24}-2 q^{22}-q^{21}-q^{20}+q^{19}+3 q^{18}+q^{17}-2 q^{16}-3 q^{15}-2 q^{14}+2 q^{13}+4 q^{12}+2 q^{11}-2 q^{10}-4 q^9-2 q^8+2 q^7+5 q^6+2 q^5-2 q^4-5 q^3-2 q^2+2 q+5+2 q^{-1} -2 q^{-2} -5 q^{-3} -2 q^{-4} +2 q^{-5} +5 q^{-6} +2 q^{-7} -2 q^{-8} -4 q^{-9} -2 q^{-10} +2 q^{-11} +4 q^{-12} +2 q^{-13} -2 q^{-14} -3 q^{-15} -2 q^{-16} + q^{-17} +3 q^{-18} + q^{-19} - q^{-20} - q^{-21} -2 q^{-22} +2 q^{-24} + q^{-25} - q^{-28} - q^{-29} + q^{-30} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{42}-q^{41}-q^{40}+q^{37}+3 q^{35}-q^{34}-2 q^{33}-q^{32}-q^{31}+6 q^{28}-q^{27}-2 q^{26}-2 q^{25}-2 q^{24}-q^{23}+9 q^{21}-2 q^{19}-3 q^{18}-3 q^{17}-2 q^{16}+11 q^{14}-2 q^{12}-4 q^{11}-4 q^{10}-2 q^9+12 q^7-2 q^5-4 q^4-4 q^3-2 q^2+13-2 q^{-2} -4 q^{-3} -4 q^{-4} -2 q^{-5} +12 q^{-7} -2 q^{-9} -4 q^{-10} -4 q^{-11} -2 q^{-12} +11 q^{-14} -2 q^{-16} -3 q^{-17} -3 q^{-18} -2 q^{-19} +9 q^{-21} - q^{-23} -2 q^{-24} -2 q^{-25} -2 q^{-26} - q^{-27} +6 q^{-28} - q^{-31} - q^{-32} -2 q^{-33} - q^{-34} +3 q^{-35} + q^{-37} - q^{-40} - q^{-41} + q^{-42} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{56}-q^{55}-q^{54}+q^{51}+q^{49}+2 q^{48}-q^{47}-2 q^{46}-q^{45}-2 q^{44}+q^{43}+q^{41}+5 q^{40}-2 q^{38}-2 q^{37}-4 q^{36}+2 q^{33}+7 q^{32}+q^{31}-q^{30}-2 q^{29}-7 q^{28}-2 q^{27}+2 q^{25}+9 q^{24}+2 q^{23}-3 q^{21}-9 q^{20}-3 q^{19}+3 q^{17}+10 q^{16}+3 q^{15}-3 q^{13}-10 q^{12}-3 q^{11}+3 q^9+11 q^8+3 q^7-3 q^5-11 q^4-3 q^3+3 q+11+3 q^{-1} -3 q^{-3} -11 q^{-4} -3 q^{-5} +3 q^{-7} +11 q^{-8} +3 q^{-9} -3 q^{-11} -10 q^{-12} -3 q^{-13} +3 q^{-15} +10 q^{-16} +3 q^{-17} -3 q^{-19} -9 q^{-20} -3 q^{-21} +2 q^{-23} +9 q^{-24} +2 q^{-25} -2 q^{-27} -7 q^{-28} -2 q^{-29} - q^{-30} + q^{-31} +7 q^{-32} +2 q^{-33} -4 q^{-36} -2 q^{-37} -2 q^{-38} +5 q^{-40} + q^{-41} + q^{-43} -2 q^{-44} - q^{-45} -2 q^{-46} - q^{-47} +2 q^{-48} + q^{-49} + q^{-51} - q^{-54} - q^{-55} + q^{-56} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

3_1 5_1