# 4 1

## Contents

 (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's page at the original Knot Atlas! 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]

### 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}]
 Knot 4_1. A graph, knot 4_1.

### 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

### Four dimensional invariants

 Smooth 4 genus 1 Topological 4 genus 1 Concordance genus ConcordanceGenus(Knot(4,1)) Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial −t−t−1 + 3 Conway polynomial 1−z2 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 5, 0 } Jones polynomial q2 + q−2−q−q−1 + 1 HOMFLY-PT polynomial (db, data sources) a2 + a−2−z2−1 Kauffman polynomial (db, data sources) a2z2 + z2a−2−a2−a−2 + az3 + z3a−1−az−za−1 + 2z2−1 The A2 invariant q8 + q6−1 + q−6 + q−8 The G2 invariant q38 + q34−q30 + q28 + q26 + q24 + q18 + q16−q10−q4−1−q−4−q−10 + q−16 + q−18 + q−24 + q−26 + q−28−q−30 + q−34 + q−38

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {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 $-\frac{136}{3}$ $-\frac{56}{3}$ $-\frac{1231}{30}$ $\frac{142}{15}$ $-\frac{1742}{45}$ $\frac{79}{18}$ $-\frac{271}{30}$

V2,1 through V6,9 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 trqj are shown, along with their alternating sums χ (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 \
-2-1012χ
5    11
3     0
1  11 0
-1 11  0
-3     0
-51    1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −1 i = 1 r = −2 ${\mathbb Z}$ r = −1 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}$ ${\mathbb Z}$ r = 1 ${\mathbb Z}$ r = 2 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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.

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.